Rethinking Out-of-Distribution Detection and Generalization with Collective Behavior Dynamics
Authors: Zhenbin Wang
Lei Zhang
Wei Huang
Zhao Zhang
Zizhou Wang
Venue: NeurIPS 2025 poster ()
Keywords: domain shift
out-of-distribution detection
out-of-distribution generalization
transfer learning
Primary area: general_machine_learning

Abstract: Out-of-distribution (OOD) problems commonly occur when models process data with a distribution significantly deviates from the in-distribution (InD) training data. In this paper, we hypothesize that a $\textit{field}$ or $\textit{potential}$ more essential than features exists, and features are not the ultimate essence of the data but rather manifestations of them during training. we investigate OOD problems from the perspective of collective behavior dynamics. With this in mind, we first treat the output of the feature extractor as charged particles and investigate their collective behavior dynamics within a self-consistent electric field. Then, to characterize the relationship between OOD problems and dynamical equations, we introduce the $\textit{basin of attraction}$ and prove that its boundary can be represented as the zero level set of a differentiable function of the potential, $\textit{i.e.}$, the spatial integral of field. We further demonstrate that: $\textit{i)}$ InD and OOD inputs can be effectively separated based on whether they are steady state solutions for specific field conditions, enabling robust OOD detection and outperforming prior methods over three benchmarks. $\textit{ii)}$ the generalization capability correlates positively with the basin of attraction. By analyzing the dynamics of perturbations, we propose that the potential is well-characterized by a Fourier-domain form of the Poisson equation. Evaluated on six benchmark datasets, our method rivals the SoTA approaches for OOD generalization and can be seamlessly integrated with them to deliver additional gains.

Rethinking Out-of-Distribution Detection and Generalization with Collective Behavior Dynamics
Zhenbin Wang's Lei Zhang Wei Huang', Zhao Zhang', Zizhou Wang* Machine Intelligence Laboratory, Sichuan University, Chengdu, China TTianuu Jincheng Laboratory, Chengdu, China
Institute of High Performance Computing, A A Singapore wangzhenbin@star n@stu scu. edu.en. lei seizhang@scu edu. cn
Abstract
(OOD) problems commonly occur when models process data with a distribution significantly deviates from the in-distribution (InD) training data. In this paper, we hypothesize that a field or potential more essential than features exists, and features are not the ultimate essence of the data but rather manifestations of them during training. With this in mind, we first treat the output of the feature extractor as charged particles and investigate their collective behavior dynamics within self-consistent electric field. Then, to characterize the relationship between OOD problems and dynamical equations, we introduce the basin of attraction and prove that its boundary can be represented as the zero level set of a differentiable function of the potential, i.e., the spatial integral of field. We further demonstrate that: InD and OOD inputs be effectively separated based whether they are steady state solutions for specific field conditions, enabling robust OOD detection and outperforming prior methods over three benchmarks. fi) the generalization capability correlates positively with the hasin of attraction. By analyzing the dynamics of perturbations, we propose that the potential is well-characterized by a Fourier-domain form of the Poisson equation. Evaluated on six benchmark datasets, our method rivals the SoTA approaches for OOD generalization and can be seamlessly integrated with them to deliver additional gains. The code is available https:://// /ttpt/gtttttt//ee com/ wwngzbb/cc
Introduction
Machine learning systems [15, 25, 70] have achieved significant success under the i.i.d. assumption, where training and test data are drawn from the same distribution, known as in-distribution (InD). However, this assumption is often violated in real-world scenarios [89, 95], as test data frequently originate from different distributions, referred to as out-of cut-of-distribution (OOD) data, which causes distribution shift and substantial performance degradation [20, 61, 21, 87]. Prior work on addressing OOD challenges primarily follows directions: OOD detection [19. 27, 49], which aims to identify test inputs that deviate from the training distribution to enhance model robustness and safety on unknown or anomalous data; 2) OOD generalization [8, 57, I], which seeks to improve model performance on OOD data to enhance model transferability and adaptability. Despite extensive prior works and diverse methodologies, recent study [72, 36] in the community inspired by other disciplines is expected to provide new perspectives and insights into OOD problems.
Recent advancements witness the success of physics-inspired deep generative models, such as the Poisson flow model derived from electrostatics [92, 93] and the model based on the stochastic reversed heat equation [63]. The former abstracts data as charges and generates samples by evolving them
The corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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High-Level Features
Initial State
Basin of Attraction Figure 1: Conceptual illustration of our collective behavior dynamics insight for addressing OOD problems. High-level features are viewed as interacting particles within basin of attraction, transition- ing from initial to steady states under the influence of self-consistent electric field.
Steady State
Electric Field
along electric field lines in augmented space, while the latter interprets the solution of the forward heat equation with added constant noise as a variational approximation to a Liffusion-based latent variable model. Overall, these approaches focus learning the field of data using physical partial differential equations (PDE), rather than directly modeling the data distribution. These impressive achievements make us realize the fundamental nature of field or potential, with data are merely external manifestations or inherent properties of these deeper underlying concepts. This motivates us model InD data using physical PDE. However, directly transferring prior work OOD problems is infeasible, as generative models primarily focus on the behavior of individual particles within field. In contrast, OOD problem center on determining field potential boundaries, which requires analyzing collective behavior rather than individual behaviors. This discrepancy motivates us to first pose the following question:
Can we find concise, theoretically grounded yet tractable PDE to model the collective behavior of the high-level features and measure the boundary of the basin of attraction?
In pursuit of this, we introduce the Vlasov-Poisson system [67, 18], dynamic physics model that describes the evolution collective particle behavior under self-consistent electric field. This system captures the feedback loop where particle positions induce changes in the field, which, in turn, influence particle positions, driving the system towards a steady state'. In this sence, we abstract high-level features from the feature extractor charged particles, with their values corresponding to initial positions and assigned uniform initial velocitics. Under the assumption that all InD features correspond to steady-state solution of the Vlasov- evolve toward that equilibrium. The region formed by such initial states is referred to as the basin of attraction (as depicted in Figure 1). For OOD detection, provide two principles: i) Vlasov- Poisson systems modeled by collective particles with distinct initial positions but identical initial velocities, typically yield unique steady-state solutions (Theorem 2.1), and ii) minor perturbations outside the basin of attraction amplify during system evolution, sharpening gradients at the boundary (Corollary 2.3). Thus, the basin of attraction is ideal for distinguishing InD and OOD features. Since this process operates on pretrained high-level features, it is therefore a post-hoc detection method. On the other hand, for OOD generalization, considering potential latent defects outside the basin of attraction, we assume that network robustness improves as the basin's 's range expands. So we next seek address the following question:
system, their dynamics will naturally
How can we enlarge basin of attraction's 's boundary to enhance model generalization?
To answer this, we decompose complex perturbations into super-
Steady state Solusion
bistributtion Field...
positions of simple plane waves and examine their collective behavior in the "lasov-Poisson system by giving dispersion re- lation that characterizes the stability of the wave modes. Through this lens, all wave modes converge to a common steady state, the system ultimately stabilizes at that solution, regardless of the initial perturbation, as illustrated in Figure Therefore, by requiring the potential in the random single-wave mode to satisfy the frequency form of Poisson equation, we expand the basin of attraction to include the perturbation points. We present Theorem 2.4 to support
Time
Figure 2: all plane-wave modes reach steady state, cross- frequency interactions dimin- ish, leading to equilibrium.
steady state refers to a condition where system's macroscopic properties remain constant over time, indicating dynamic equilibrium between particle motion and the self-consistent field. A wave with uniform amplitude and phase across infinite, parallel wavefronts, propagating fixed direction.

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2 (x)
Uncertainty Estimation: Df do, So. F6 (1 -
Fole, v)
Statiomany bisti
|| V. ()) v) vov(")) VV./"")-v.PP6)e.. v..6(e'.x)])
Electric
Qul Electrie Pizeeniial
reb
Force
Model Evaluation:
Non-Steady
G for f6) = Im(w), s.t.
falz)
State
k V.Fo(a.v) do
geverabzation
Steady State
w w-k-v
Figure 3: Illustration of our CBD-based framework. CBD is derived from the Vlasov-Poisson system and models the dynamics of high-level features under an electric field. The basin of attraction quantifies the region where features from the InD training set X spontaneously converge to steady state. Based on CBD, we propose two methods targeting OOD detection and OOD generalization.
this design and discuss its impact on the model's generalization ability, building on Theorem 2.5 and Corollary 2.6.
In summary, the contributions of this work are three-fold: i) Theoretically, we introduce an application of fields and potentials in OOD problems from two novel perspectives: the collective behavior dynamics of high-level features and complex perturbations under self-consistent electric fields, with supporting theories. ii) Methodologically, we propose physics-inspire Collective Behavior Dynamics (CBD) framework for OOD problems. It is simple, effective, and compatible with other methods. ii)) Experimentally, our method outperforms matches SoTA approaches across three OOD detection benchmarks and six OOD generalization benchmarks, and can be integrated with them for further improvement.
Prior Arts. We discuss related work and defer a concentrated account to Appendix A.
2 CBD: Collective Behavior Dynamics
Let X {(x,y)} denote the training set, consisting of i.i.d. samples drawn from the joint distribution P = X' x Y/, where X' Rdd represents the input space and y = (1,2, C} is the output space. Consider a classification network parameterized by 0, where the network maps inputs x E X' to probability distribution over the output space V. We illustrate our CBD-based approaches Figure 3. In the following, we first present the formulation of the Vlasor-Poisson system (Section 2.1), and then introduce the details of basin of attraction (Section 2.2). followed by its application OOD detection. Finally, we formulate the dispersion relation (Section 2.3) and demonstrate how CBD can be utilized for OOD generalization.
2.1 Formulation of Vlasov-Poisson System
The performance of classification networks typically relies on high-level feature extracted from individual inputs. However, this overlooks the deep physical constraints that exist between high- level features across InD data. To model the behavior of high-level features z=fo (x). where fo(-) denotes a feature extractor, we treat these features extracted from InD data as charged particles in the space R2, Due Coulomb interactions, these particles spontaneously move in space. In the absence of collisions and external magnetic fields, their dynamics can be effectively described by the Vlasov-Poisson system, which consists of two fundamental equations: the Vlasov equation and the Poisson equation. Specifically, the Vlasov equation is given by:
OF OF v,t) +V V2F(z, v) + E(z,t) F() Øt
(1)
(1) preserves the Hamiltonian structure and energy conservation of the system, where Fiz, v, e x xTR +R+ is the particle distribution function time i, defining the particle density position z and velocity v F(z, ?T? corresponds to the electric field generated by particle distribution (assume all particles have the same charge). Vz and V, represent the gradient with respect to position and velocity, respectively. (1) describes macroscopic space-time changes.
The Poisson equation characterizes the relationship between the electric field E(z, t) and the particle distribution F(z, (), and can be expressed as:
V お 2 1) = - / F(z, v, 1jdv (iion (z)), E(z, 1) (z)
60
Here db 1): x denotes the electric potential, f F(2, v, f)dv the charge density (i.e., the total charge position 2), and eg.00 the permittivity constant. Pion (z) denotes lon background
3

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density, assumed to be a spatially uniform constant. It serves to neutralize the system and ensure potential solution. (q.( and (2) together describe particle dynamics under the electric field.
Theorem 2.1. Consider N Plasov-Poisson systems with steady state distributions F((z. v) =
n,(z) 8(v vo). where each spatial density (z(z)) E (x22)) n L (R2) represents the spatial density for the i-th system, and L (R) and (R) denote respectively the spaces of integrable and essentially bounded functions on R. The Dirac delta distribution ô(v vv) is centered a
v -Vu, indicating a velocity distribution concentrated at that point. Assume that each n. nz(zz) satisfies Vo. (z),(z) = O and that the ion background density is given by Pion, = (z) If n. (z) + ny (z)
for + j. then the solutions F; are distinct.
We prove that in different multiple systems, where particles share identical mass and initial velocity af =0) are distinet. Vo but have distinct spatial density distributions, the steady state solutions (i.e., Øt Using the provided Vlasov-Poisse equations, a neutralizing ion background, and a density gradient condition, we establish the result via distribution theory. The proof is deferred to the Appendix B.1.
2.2 Basin of Attraction and Application to OOD Detection
Building upon the concept of the Vlasov-Poisson system, we introduce the notion of basin of attrac- tion to characterize the system's 's dynamical behavior under both InD and OOD inputs. Let F (z. v) de- note a steady state solution to the -oisoon system. We define the basin of attraction B(F*) as the set of initial distributions Fo such that the time- solution asymptotically approaches F":
B(F*) foo E F lim F(t, F in some topology).
Here, F denotes the admissible function space (e.g., Sobolev or Wasserstein space). This asymptotic behavior can be equivalently characterized by the steady state form of the Vlasov-Poisson system,
V2F* + E² F.F' (z.v) = 0, (z)
Ved (lmll) Pion (2)).
(fF
V (time-independent)) (4)
Definition 2.2. Under thermodynamic equilibrium, the steady-state electric potential x² satisfies the nonlinear Poisson-Boltamune Boltzmann equation:
V20 6" (z) (e (e-6"(e) Pion (z)).
(5)
(5) describes the equilibrium electrostatic potential resulting from Boltzmann-distriuuted charge density balanced against a fixed ion background. The derivation is provided in Appendix B.2.
Corollary 2.3. Consider d (z) be the steady state electric potential satisfying the nonlinear Pois- SontBoltemnnn equation. Then the basin of attraction's is boundary 0B can be implicitly represented as the zero level set of a scalar functional over (²(z) (z),, i..,
0B = f (z) - -0.
(6)
where G is differentiable function that depends on system configuration, such as energy thresholds, local potential curvature, or external field geometry. This representation enables differentiable modeling and learning of basin boundaries via neural function approximators.
Corollary 2.3 establishes that the basin of attraction B associated with the steady state dynamics is fully characterized by the solu- tion d² (z) to the nonlinear Coisson-Boltzmann equation, through a differentiable level-set condition ( (z))-O. It implies that the geometric and lopological structure of B. including its bound-
Electric Potential
Basin of Attraction's Boundary
InD
ary 0B, is encoded in the electric potential d² We defer the formal Appendix B.3. Given the challenges of directly solv-
Electric Field
OOD
ing high-dimensional nonlinear PDE of E4.(4) and the need for framework compatibility, we propose incorporating two parallel
MLP branches following the feature extractor: Fg for predicting the steady state distribution function F* and de for estimating the steady-state state potential Ø², both running in parallel with the linear classifier So. As shown Figure 3. This architecture ensures CBD compatibility with most existing methods. Then we formu- late a physics-infee informed loss, which enforces the consistency of the learned potential with the steady state field equation. Because
Figure 4: Electric field, poten- tial, and basin of attraction's 's boundary. Particles inside con- verge to steady state; those out- side behave chaotically.

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Poicutial Field
Frequency Spectrem
Basin Boundary
Figure 5: Effect of dispersion relation regularization on the Iearned potential field, its frequency spectrum, and the recovered basin of attraction. Left column: Predicted potential fields with (left) and without (right) dispersion relation loss. The regularized solution is smoother and more physically consistent. Middle column: Corresponding frequency spectrum. Dispersion relation regularization suppresses spurious high-frequency components, aligning the spectral structure with physical expecta- tions. Right column: Estimated basin boundaries (contour of Φo (z) = Dispersion regularization leads to a wider and smoother basin, indicating improved global consistency and robustness.
the Ion background density Pion (z)n(Z) is constant, it introduces only a uniform offset and can be safely omitted from derivation, Thus the resulting Poisson loss is
1
Coossoon (z; fo. oo)
V2 Φz())
+ _ -0
60
The steady state potential 6 encapsulates the overall geometric and topologiall structure of the InD data, while also satisfying the Vlasov equation. So we get the following Vlasov loss:
L (z; So, $0, Fo) = ||v Fo v) (Φo(2) Fo v)ll"
(8)
Then we leverage Cpoisson and Lviasov as a physically grounded uncertainty score to identify whether z" fxx)(x") lies inside the basin of attraction or corresponds to OOD behavior,
InD if Pooisoon (z': fo, (e),c)) do) and L (z'; for 0g Fo) Fo)
D(z" N0, do, Fo,, X) OOD otherwise
(9)
where 01 and >200 is thresholds and x* is test data. This approach not only offers physical interpretability, with the score directly quantifyyng deviation from the governing PDE. But also geometric alignment with basin structure, as Corollary 2.3 demonstrates that (z)" exhibits sharp gradient near the boundary OB, enabling robust OOD detection without statistical estimators.
2.3 Dispersion Relation and Application to OOD Generalization
OOD data's high-level features may fall outside the basin of attraction of InD data, preventing spontaneous transition from initial steady states under given electric fields. From this perspective, we hypothesize that network robustness is proportional to the range of the basin of attraction.As the basin of attraction contracts, larger activity spaces emerge, increasing the probability of potential defects. As illustrated in the second image of the right column in Figure 5, without additional regular- ization, the learned boundary of the basin attraction for exhibits fragmentation, demonstrating not only sensitivity to input data but also diminished area (green area). Enlarge this basin implies that the system can accommodate larger initial deviations, without diverging or converging to alternative equilibria, thereby improving the robustness of the steady state.
We define the set of initial perturbations (Jo, th (F===(F-F* -- To investigate the collective behavior dynamics of perturbations, we employ plane waves for decomposition. We define the plane wave as (2( (k= and (z, v, f))
where K denotes the wave vector specifying the spatial frequency and direction (we treat k as random vector hyperparameter in our experiments), and no=wit-i is the complex frequency. The scalar v and f((v represents the mode amplitude. This representation enables any perturbation to be expressed as a superposition of decoupled modes, and in systems exhibiting translational invariance, plane waves naturally arise as cigenfunctions. transforming PDE into algebraic forms. Substituting v and f into E4,(4), we get the perturbation collective behavior dynamics equations:
0. t) + Vzf(z, t) ('v(2,t) = - f(, dv.
(z) V.f(2, t) (V.F*(z, v)) (Vab(z,))) ()),
(10)
Substituting the plane wave into (10), yielding
{ -iwf(v) ik vf(v) + (- (z) v.f(v)) ((V,FF(Z, ik, f(v)dv.
(11)
| 12 = cn
5

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Assuming a homogeneous steady state for simplicity (i.e, (z) is constant and v=) the electric field E (z)=0, and (q.(1 simplifies to:
f(v + ik vf(v) =VVV ik.
(12)
Solving for f(v), we obtain:
3 (V.
f(v)
(13)
w k
Integrating over velocity space and substituting into the Poisson equation, we get:
/
k. 1 k (v) dv dv 0.
kww
(14)
60
The solutions of D(w, k) is (k)= -als which determine the stability of each mode. To assess stabilityy we define energy-like liie quantity as the squared magnitude of the perturbation, (v( t) (or alternatively [f(z, v. t)l? with analogous derivation steps following). Substituting (k))=w, + iws, (z. this becomes (z, =)|2=| 2 since the exponential term e ((xx-(w,tia = (wx iw ewit
-Wws,
and the magnitudes of the complex phases ed and e -,e, are unity. Thus, when w,>0. the factor ewutt grows exponentially, causing the energy-like quantity to increase, potentially destabilizing the system and contracting the basin of attraction. Conversely, when w:<0, e: decays exponentially, reducing the energy-like quantity and facilitating recovery the steady state. In the nonlinear regime, perturbations decompose into multiple modes, and decaying modes (w;<0) can mitigate the growth of unstable modes (w;>0) through modal interactions, such energy redistribution. Therefore, the imaginary part of w can be used to evaluate the generalization ability of the model.
To enlarge the basin of attraction, we aim to suppress perturbations associated with positive growth rates, i.e., those satisfying wi > as they contribute to instability and hinder recovery to equilibrium. To this end, we introduce the following dispersion relation loss that regularizes the consistency between the electric potential and the charge distribution in the frequency domain
Laiis( (2; Sos sbo) = Exll
|k| (% + (2)
(15)
S.I. (z (z(ze) -ke-
(z( e -du(x) l-
This loss arises from the Fourier-domain formulation of the Poisson equation. Minimizing Eq. therefore enforces the constraint across sampled wavevectors k (see Figure 2). From a dynamical perspective, satisfying this balance prevents the system from developing unstable frequency compo- nents, which would otherwise drive the ((w, k0 toward solutions with w;>0, such imbalances act as implicit forcing terms that require growing temporal responses. In contrast, minimizing the dispersion relation loss encourages configurations where w,<0, ensuring exponential decay of perturbation energy and improved return to steady state. This regularization thus enhances stability and expands the basin of attraction. Figure 5 presents a comparative analysis of predicted potential fields, frequency spectra, and estimated basin boundaries before (right) and after (left) incorporating the dispersion relation loss, demonstrating its superior efficacy in enlarging the basin of attraction. According to (15() the underlying theoretical rationale can be formulated as follows.
Theorem 2.4. Define the approximate basin of attractor basin as the residual sublevel set
Bg = fz (V (z())) + - < A}.
(16)
6o
Then, under mild regularity assumptions, increasing the spectral regularization strength > leads to a strictly larger basin measure:
mcas (5) (2000020)) 2 mcas (B) (a()am=0)) 0)
(17)
Theorem 2.5. Let dg. (R (RR )R2), respectively, with 8> Z/2, approximate the true
solutions 6", F of the Vlasov-Poisson system. Training data (z, v)D are i.i.d. from 2xR2 The generalization error is: =)) sion of Littpp reduces Egon(() by constraining the Sobolev norm of coe satisfying:
(z)= (x)) (z))F=+| ())F"(z, v)l.]. The inclu-
E Ex (aaup] 2 cllool1?ra-- -C
(18)
for constants C > 0, thereby reducing the model complexity term in the generalization error bound.

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Table I: OOD detection performance CIFAR-10 and 10F benchmarks, with the top three results highlighted. Complete results for all baseline methods are provided in Appendix E.
MINIST
SSVHN
Textures
Places665
Average
Method
FPR95.
AUROCT
FPRO5.
AUROCT
FPRO5L
AUROCT
FPR95,
AUROCT
FPR95L
AUROCT
CIFAR-IO Benchmark
OpenMax
23,33 40
90.50-144
15.40-14 12
89.77
31.50s 4s
89.58
38.52 121
88.63+
29.69. 121
89.62 111
ODIN
1333
95.24
68.61.
84,58 14
67.70:
16.94
70.36.
85.07
57.62
87.96.141
MDSEas
1.30-a
-9.17-44
76.07
77.40zum
94.16-sw
52.47
61.47..m
73.90=
RMDS
21 1m
93.222115 tm
23 14
91 84
25 28g
92. 25
51,20 km
91 a1
25.35
92.20
Gram
70,30=
72.64g74
33. 13.91211
91 .4
94 11
62.334
90,49_-1
60 00.44x.0
62.34
71.73=
RoAct
92.81 18.77..10
50,23:
$1.42a
44.20:133
44.90
90.42:141
VIM
18.36-1.
1M
19:292291
94 S0ggns
21. 21:14:11
95. 1520s
41,492211
89 m9
25.05:
93.482 174
KNN
20.05 1.2
91.26 428
22. 72.60_1 1:3
92.67 o23
24.06 ea5
93.161
50.38. 1:53
91,771
24.27 6.4
92.96..14 114
ASH
70.00-11.:
-3
63.64-
73.46
84,59
17.45+
79.89, -7.89-13
79.03.
78.49
SHE
12.22 20.99
476
62.74
86.38:
:4.60
11
76.36 5:32
68.48 sas
85.32
GEN
23.00 1/5
23
14
91.77
40. c.
90. 14:44
47,03= km
89, 19.46
34.73z 194
91.35= 2.
NAC-UE
15.14- 24
94.86 Lis
14.33 124
96.05
17.03
95.64..o
26.73- 1:0
91 P1.85 oza
18.31 6x2
94.60 1.0
CHD-De
15.20
97.48
14.32
97.85
18.44
97.03
22.75
93.18
g7.65
96.39.mm
CIFAR-100 -FFOR-100BB Benchmark
Max
53.82=
76.01
53.20 17
82.072
56, 16.12s
80.56g
54 $4.83=
79 29
$4,50 es
79,48: 1:
ODIN
45:94_13
67.41:
62.37
79.33:
59.71=
79.45..23
28
MDSSEs
2.
2M
53.76218
84 04
69.7521
26.61
66. (6.74:11 15s
06.001 1M
RMDS
52.05 123
79.74 245
$1.65 306
84.89 L3
$3.99 us
83.65
53.57
14
83,401
52.81 eas
82.92 1-3
Gram
53.53_11
80.71
20:06:115
95. 55
19
70.79213:
94.67=1
46 18
64,44: 137
73.36=
ReAct
56.04
1.s
50.41
83.01
15.04
80
14
a0
54,20 135
VIM
48.32 um
46.22 sis
1.
85.914mm
61.57=*m km
50.74: 191
81,70= xm
KNN
48.58 40
82 36.6-.11 1:3
51.75 A12
84.15
$3.56 L2e
60.70 1M
79 am
53.65 0.21
82.40= 111
ASH
66.5K_11
46:00
85 0..00111
N0. N0.72
62.95_1
59.211
80.58=11
SHE
58.78- am
76.76 un
80.97
73.64
65.24 k.
a6
64.12;
76.92=
GEN
53.922 xm
78. 78.29-2mm am
55.45 2T
81 11
61.23g
78. 04
$6,25 1:1
80 987
56.71: 199
79.60 0.68811
NAC-UE
21.97.
93.15
24.39
92.40
40.65
89.32
73.57
73.05
40.14:
86.96
CBD-De
27.93
96.29
18.72
95.80.2m
37.34.236
90.02.1.
55.23300s
81.50
90.90anas
Corollary 2.6. Under the same assumptions with Theorem 2.5, let H= {(vo, Fo) :0 E Q be the hypothesis class. The generalization bound is:
Egen(t) S Emaii(()) + 2Rad(H) + O O(Vloog(//6/mm)
(19)
where (H) is the Rademacher complexity, and is the sample size. The inclusion of Lassp reduces Rad(H) ()) by restricting ôe to a subset Hd C 112, tightening the generalization bound.
Theorem 2.4 proves that incorporating the loss Edisp enlarges the basin of attraction. Theorem 2.5 further establishes that this loss term decreases the generalization error, and Corollary 2.6 presents the resulting tightening generalization bound. Proofs are deferred to Appendix B.4, B.5 and B.6. For OOD detection, the loss is defined as LD - L + Epoisuum For OOD generalization, we enhance the training by incorporating Lus along with the standard classification loss [81], yielding the overall loss LGan e CCasssity +a (Lvlasov + Possson + 3BLdsp. which ensures robust performance on the primary task. For detailed pseudocode, refer to Appendix C.
3 Experiments
Steup. For OOD detection, following the latest OpenOOD [94], we evaluated our CBD-De on three InD datasets: CIFAR-10, CIFAR-100) [38], and ImageNet-Ik Ik [12]. For CIFAR- and CIFAR-100, we employed ResNet-18 18 and assessed performance against four OOD datasets: MNIST [13], SVHN [59], Textures [10], and Places365 [101]. For ImageNet-1k, 1k, we utilized pre-trained ResNet-50 and Vit-b16 models, evaluating against three OOD datasets: iNaturalist [79], Textures, and
[84]. We compared our approach with 22 SoTA OOD detection methods, with comprehensive results provided in Appendix E. For OOD generalization, we adhere to the Domainbed [23] in our experimental setup and use BCE as classification loss, evaluating our CBD-Gen on five datasets: VLCS [16], PACS [45], Offi ceHome [82], Terra Tnc [5], and Doma nNet [60]. For each dataset, we report the leave-one-out test accuracy, averaged across splits where one domain serves as the test set and the remaining domains as the training set. Our method is compared against 18 SoTA baselines, with comprehensive results available in Appendix F.
Metrics. For OOD detection, we employed two threshold-fit evaluation metrics, consistent with established conventions: i) FPR95, the false-posiin dodee-oostivee-rte of OOD data when the rrsetvsiiive--rtee of ID data is at 95%, and ii) AUROC, the area under the receiver operating characteristic curve. For OOD generalization, we adhere to the standard practice of using accuracy the evaluation metric.
Implementation Details. In the interest of brevity, we defer details such as training procedures, hyperparameter configurations, and baseline methodologies to Appendix D.
7

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Table 2: Performance of OOD detection (AUROC) ImageNet- Ik, with full results in Appendix E.
Dataset
Backbone
OpenMax
MDS
RMDS
ReAct
VIM
KNN
ASH
SHE
GEN
NAC-UE
CBD-De
ResNet-50
92.05
63.67
87.24
96.34
89.56
86.41
97.07
92.65
92.44
96.52
97.05
iNaturalist
Vit.bl6
94.93
96.01
96.10
86.11
95.72
91.46
50.62
91.57
93.54
93.72
97.26
Average
93.49
79.84
91.67
91.23
92.64
88.94
73.85
93.11
92.99
95.12
97.16
R5sNet-50
87.62
69.27
85,84
91.87
90.30
87.04
93.26
86.52
89.26
91.45
92.65
Opealmago-O
Vit-b66
87.36
92.38
92.32
84.29
92.18
89.86
55.51
91.04
90,27
91.58
91.89
Average
87.49
80.83
89.08
88.08
91.34
88.45
74.39
88.78 78
89.77
91.52
92.27
ResNet-50
88.
89.80
86.08
92.79
97.97
97.09
96,90
60
87,59
97.90
98.31
Textures
Vit
85.52
89.41
89.38
86.66
90.61
91.12
48.53
92.65
90.23
94.17
94,30
Average
86.81
89.61
87.73
89.73
94.29
94.1 11
72.72
88.91
96.04
96.31
Table 3: OOD generalization performance on DomainBed, with complete results in Appendix F.
Algorithm
Mixstyle
MMD
ARM
MIL
MLDG
Mixup
CORAL
SAM
SAGM
OGA
C'DD-Gen
PACS
85.2g0
84.7ams
85.
84.6
14
86.2
85.8
86.6gs
87.3
87.7
VLCS
77.9
77.6
77.2
77.2
77.4
78.8
79.4
80.0
79.9.
80.5
OfheHHome
60.4maa
66.3
64.8
66.4
66.8
68.1
68.7a13
69.6
70.1
70.1-11
71.4-13
Terraine
44.0aMa
42.2
45.3
45 65.6212
47.7
47.9am
48.6
43,3
48.8
50.6
50.5
Arg.
66.9
67.7
68.5
69.2
69.5
20.3
69.3
71.4
21.7
72.5
DomainNet
34.0am1
23,4
35,5
40.6
41 22.2
40.3am
41.5
44, 44.3
45.0
45.2
45.9
Total
60.3
58.8
61.7
62.9
63.6
63.4
64..5
64.5
66.1
66.3
67.2
Quantitative results on OOD detection. Ta-
Table 4: Evaluation of Gen'S compatibility with advanced OOD generalization methods.
bles and 2 present the results of our CBD-De on the CIFAR and ImageNet datasets, com- pared against 22 SoTA methods. Confidence in-
Methods
PACS
VLCS
OfficeHooe
Terralic
Arg.
baseline
85.8
79.4
09.6
43.3
69.3
tervals in the tables reflect mean + standard er-
SAM
with CoD
71.7
51.0
73.0
(+2.6)
(+1.3)
(+2.1)
(47.7)
(43.7)
ror over three independent test runs. Our CBD- De consistently outperforms all SoTA meth- ods in average performance, achieving record-
baseline
%6.6
80.0
20.1
48.8
71.4
SAGM
with CBD-Gen
88.2
80.7
71.5
508
72.8
&
(+1.6)
(40.7)
(+1.4)
(42.0)
(+1.4)
breaking results across three benchmarks. Specifically, on CIFAR-10 and CIFAR-100, CBD-De reduces the FPR95 by 0.66% and 5.33%, respectively, and improves the AUROC by 1.79% and 3.92% compared to the most competitive baseline [52]. On the large-scale ImageNet-1k dataset, CBD-De enhances AUROC scores across various backbones and OOD datasets, with average AUROC improvements of 2.04%, 0.75%, and 0.27% over the strongest competitor.
Quantitative results on OOD generalization. Table 3 reports the average OOD accuracies of SoTA generalization methods across five benchmarks. For fair comparison, all methods are trained on the same amount of data. While additional data augmentations could further improve performance, they are not applied to isolate the effect of the core methods. The average performance of CBD-Gen across all evaluated datasets surpasses that of the most competitive benchmark by 0.9%. Specifically, on the large-scale DomainNet dataset, comprising 586,575 images across six domains, CBD-Gen achieves a 0.7% improvement over the leading method. Furthermore, CBD-Gen significantly outperforms the current SoTA on the OfficeHome dataset. We attribute this to its use of electric field modeling, which enhances the network's 's ability to capture discriminative features from limited data. In particular, the electric potential exhibits sharp gradient transitions near the basin of attraction's 's boundary, leading to larger inter-class margins. This effect is beneficial for OfficeHome, where each class contains only ~ 240 samples on average, and substantially fewer than the 1,400+ samples per class available in other datasets, highlighting CBD-Gen's 's strength in low-data regimes.
Compatibility results. We further investigated the potential of CBD-Gen by integrating it with two advanced algorithms: SAM [17] and SAGM [86]. During training, we incorporated two MLP branches after the feature extractor and incorporate additional Lpoison: L-Vlasov and Laisp losses into the existing algorithmic frameworks. These branches are removed during testing. Results are shown in Table 4. Notably, CBD-Gen maintains consistent and superior performance across multiple datasets when combined with other methods, further validating its strong generalizability and compatibility. This demonstrates that CBD-Gen is not only effective within traditional training paradigms but can also function as an enhancement module that integrates seamlessly with various advanced methods, exhibiting extensive adaptation capabilities.
Text OOD Detection. To demonstrate the modality-agnostt nature of our post-hoe OOD detection method CBD-De, we further conduct experiments on text classification benchmarks. Following the evaluation protocol established in [4], we evaluate our method on standard text OOD detection datascts. Specifically, we adopt the widely used SST-2 [71], IMDB [54], and Yclp [99] datasets,
8

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Table Performance of OOD detection (AUROC) on text data.
ID
OOD
MSP
Energy
GradNorm
KLM
ReAct
DICE
KNN
VIM
CBD-De
SST-2
IMDB
83.2
82.7a22
70.3
55.0-17
83.3
34.5-107
87.2+17
83.9a33
89.0
SST-2
Yelp
75.7
75.0zx1
61.3+27
51.3-10
75.7
35.4
80.
89.3
Yelp
IMDB
79.5
79.2#16
71.7a19
38.6-13
79.5a15
26.8+51
84.7s0s
88.6
90.2:
Mixstyle
SAGM
Ours
Mixstyle
SAGM
Ours
08
Ideal
0.0
Ideal
Ideal
Accuracy
Gap
Accuracy
Gap
Accuracy
Gap
0.5
02
86
25
0.0
05
62
24
5.0
03
02
10
Confidence
Confidence
Confidence
DOD Accuracy
0OD Accuracy
3.5
0.05
0OD
0.01
02
0.1
0.001
02
Hyperparameter Comparison of
Hyperparameter Comparison of
530
1030
3320
2300
2306
5203
300
1000
3500
3000
2500
3000
33
20
30
Step
Step
Batch Size
Figure Confidence calibration comparison and hyperparameter analysis on the OfficeHome dataset.
where each dataset alternates as InD and OOD data. As shown in Table 4, our method achieves competitive performance compared established baselines. These results confirm that CBD-De, though originally designed for visual OOD detection, generalizes effectively to text modalities without requiring any modality adaptations.
Confidence calibration analysis. We evaluate the confidence calibration performance of Mixstyle [102], SAGM [86], and our proposed CBD-Gen on the OfficeHome dataset, with re- sults visualized via reliability histograms in Figure (upper row). Predictions are divided into bins based on confidence levels, and the mean accuracy is calculated for each bin. An optimally calibrated model should exhibit diagonal pattern in the reliability diagram, highlighted by the gray bars. We observe that Mixstyle exhibits poor calibration in low confidence intervals (0 to 0.5), likely due to these data augmentation-bant significant distribution shifts. In contrast, while SAGM outperforms Mixstyle in these low intervals, underperforms mid- confoiddeeeeee intervals (0.5 to 0.8), possibly because these sharpness aware minimization methods overly smooths samples with moderate gradients, skewing their probabilities toward the extremes. Our proposed CBD-Gen, however, excels in both low and mid-confidence intervals. This advantage may arise from decomposing perturbations into multiple plane waves, enabling two key operations: i) perturbing random frequency components indirectly create broader OOD data, addressing calibration deficits in low-confidence regions, and ii) optimizing the basin of attraction's 's boundary, defined by a continuous PDE over InD data, ensuring smooth and consistent parameter adjustments for accurate probability estimation in mid-confidence intervals.
methods struggling to generate OOD samples with
Hyperparameter analysis. To maintain clarity, we primarily use the OfficeHome dataset to investi- gate the impact of two loss-related hyperparameters in CBD-Gen, i.e., a and B, as well as the effect of training batch size on model generalization. As shown in the bottom row of Figure 6, we first vary a while fixing 3 0.01. The results indicate that model performance remains stable over broad range of a values (0.01 to 0.1), suggesting that CBD-Gen is robust to changes in the relative weighting of Poisson and Vlasov losses within this interval. Next, fixing 0=0.01, we examine the influence of and observe that generalization performance degrades slightly at both low (3=0.001) and high (3=0.1) values. small may weaken the model's s ability to enlarge the basin of attrac- tion's 's boundary, limiting generalization, while overly large 3 could lead to excessive emphasis matching plane wave modes, at the expense of accurate electric field modeling, ultimately impairing classification performance. We also investigate the effect of batch size CBD-Gen's 's performance, based on the assumption that accurate modeling of electric fields and potentials from InD data benefits

---

from larger batches. As shown in Figure 6, general- ization performance improves with increasing batch size. Batch sizes between [32, 64] offer strong, po- tentially optimal results, though hardware constraints limit evaluation at larger scales.
Sample Points
177
AC Accuracy ocuurc
Plane wave analysis. Experimentally, the wavevector k in (15) is sampled from a Gaussian distribution k~ I²) for each batch (see pseudocode in Ap- pendix C). The scaling factor a controls the spread of sampling in the frequency space and directly af- fects the strength of dispersion relation regularization. Additionally, the wavenumber N determines the dis- cretization resolution in the frequency domain, signif- icantly impacting both the stability of regularization and computational efficiency. We conducted system- atic ablation studies on different values of a and N. As shown in Figure 7, increasing N leads to consistent generalization performance (from blue to red/orange sample points). This is attributed to a finer and more comprehensive frequency decomposition, enabling the model to better capture and adapt to various domain shifts. However, excessively large N degrades training efficiency. To balance performance and efficiency, we set N=250 the above experiments. Furthermore, comparative analysis of various scaling factors revealed that both insufficient and excessive pling ranges degrade model generalization performance. Smaller scaling factors preserve excessive high-frequency components, introducing non-physical oscillations in the potential field, while larger factors over-smooth the field, eliminating critical physical structural information. Experimental results demonstrate that a-0.5 yields optimal performance, balancing physical consistency with discriminative feature preservation.
03
100
352
Wavenumber
10
25g
Scaling Factor
13
6.5 0.e5
6765
6.70
Figure 7: Comparative analysis of wavenum- ber N and scaling factor a on OOD general- ization task using the OfficeHome dataset.
improvements
Limitations. We would like discuss several limitations identified in study. First, there is an opportunity to enhance CBD by reformulating the distribution and potential prediction heads as a multi-step PDE solver. Such design could improve numerical stability and provide more faithful approximation of the underlying dynamics, potentially leading to greater precision compared to directly solving for the steady state. However, this reformulation may introduce additional parameters, hyperparameter sensitivity, and computational overhead, which should be carefully balanced against the expected gains. We provide an illustrative experiment in the Appendix D.8 to demonstrate this limitation. Second, the CBD algorithm requires extensive storage of first- and second-order intermediate gradients, resulting in sigmificant GPU memory demands. This not only increases training costs but also restricts batch sizes and model scalability in practice. This constraint could limit its applicability to large-scale high-resolution tasks unless more memory-efficient optimization strategies are developed, such as gradient checkpointing or mixed- precision computation.
4 Conclusion
In this work, we present a physics-inspired collective behavior dynamics perspective to address the OOD problem. We conceptualize high-level features charged particles and explicitly model their collective behavior under a self-consistent electric field using a Vlasov-Poisson system of dynamical equations. By solving for InD electric potential and distribution functions, we link the InD boundary to the system's s steady-state basin of attraction, enabling effective OOD detection. Furthermore, we analyze the collective behavior dynamics of perturbations within the system and resolve the OOD generalization problem by enlarging the boundaries of the basin of attraction. Extensive experiments across diverse tasks, benchmarks demonstrate our CBD's advantages over current leading methods. Following this line of thought, we hope our perspective inspires the community to consider physical dynamical behaviors in OOD problems, which naturally aligns with principles of interpretability.
Acknowledgments
This work was supported by the National Science Fund for Distinguished Young Scholars, PR China under Grant No.62025601, Transformation Foundation of Tianfu Jincheng Laboratory (2025ZH013), National Natural Science Foundation of China under Grant U24A20341 and Sichuan Province Innovative Talent Funding Project for Postdoctoral Fellows under Grant BX202512.
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16

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Appendix
Table of Contents
Prior Arts
18
B Proof
19
B.1
Formal Proof of Theorem 2.1
19
B.2
Formal Proof of Theorem 2.2
20
B.3
Formal Proof of Corollary 2.3
22
B.4 Formal Proof of Theorem 2.4
23
B.5
Formal Proof of Theorem 2.5
26
B.6 Formal Proof of Corollary 2.6
27
Algorithm Protocol
30
D Additional Experiment Settings and Details
32
D.1 Baselines
32
D.2 Benchmarks
32
D.3 Model Architecture
34
D.3.1 OOD Detection
34
................................
D.3.2 OOD Generalization
................................
................................
35
D.4
Hyperparameters
................................
35
D.5 Experiment Infrastructure..
36
D.6 Ablation on Initial Velocity Assumption
36
D.7 Sensitivity Analysis of MLP Architecture
36
D.8 Validation of the MLP Steady-State Approximation
37
Full OOD Detetion Results
37
Full OOD Generalization Results
41
17

---

A Prior Arts
OOD detection [27] aims to identify inputs that significantly deviate from the training data distri- bution, task critical to ensuring model reliability and safety in real-world applications. Existing approaches can be broadly categorized into post-hoe and prior-based methods. Prior- based methods integrate OOD awareness into the training phase by modifying training strategies [30], loss func- tions [29, 55], or model architectures [78]. However, prior-based methods often require access to OOD samples or carefully crafted synthetic anomalies during training, which may not be feasible or generalizable across deployment scenarios. Additionally, such methods typically involve sub- stantial modifications the training pipeline or model architecture, increasing computational cost and limiting applicability to pre-trained models. In contrast, post-hoe methods operate on models already trained on InD data, making them more practical and flexible for real-world deployment. These methods rely on analyzing model responses, such as confidence scores [27, 100]. output gradients [31], or intermediate activations [42], to estimate the likelihood that an input belongs to the training distribution. Their plug-and-plan nature and compatibility with existing models make post-hoc approaches particularly appealing for scalable OOD detection. Our proposed CBD-De builds upon a pre-trained model by training two additional MLP branches with InD data, while freezing all parameters of the pre-trained model (including both feature extractor and classifier), which categorizes it as a post-hoc method.
OOD generalization [48] focuses on improving model performance on unseen distributions without requiring access to OOD data during training. It aims to learn representations that remain effective under distributional shifts, enabling better generalization to unknown environments. Existing OOD generalization approaches can be broadly categorized into three groups: data augmentation, represen- tation learning, and model regularization. Data augmentation methods [102, 11] simulate distribution shifts by applying input transformations or generating synthetic domains. Representation learning methods [1, 69, 35, 90, 88] aim to learn Nomain-invariall or momaan-dissriminatii features. Model regularization methods [33, 64] introduce constraints during optimization to improve robustness. While OOD generalization does not require OOD data, often assumes access to multiple training environments or sufficient data diversity, which may not always be available in practice. By modeling the field or potential of InD data, our approach demonstrates superior performance over existing methods on datasets with limited class samples, as substantiated by comprehensive experiments.
Vlasov-Poisson system [83] is a fundamental kinetic model describing the evolution of collisionless particle distribution under self-consistent field. Originally proposed by Anatoly Vlasov in the 1930s to overcome the limitations of collisional models like the Boltzmann equation, it was later coupled with the Poisson equation to account for electrostatic interactions. Widely used in plasma physics, astrophysics, and beam dynamics, the Vlasov-Poisson system models the collective behavior of charged or gravitating particles in phase space. Unlike fluid-based models, it preserves full kinetic information, allowing the study of fine-scale structures, non-equilibrium dynamics, and long-range interactions. Phenomena such phase mixing, filamentation, and Landau damping naturally arise in this framework [56, 67]. Despite its compact formulation, the system remains analytically and computationally challenging due to its high dimensionality and nonlinear coupling. Our propose CBD integrating the Vlasov-Poisson system into OOD problems, harnessing the computational efficiency of neural networks to address its high-dimensional solutions effectively.
Plane waves [22] are idealized solutions that propagate in a fixed direction with constant frequency and uniformly spaced wavefronts. Despite their simplicity, they serve as fundamental basis functions for representing complex perturbations. Leveraging Fourier analysis, any small perturbations can be decomposed into a superposition of plane waves with varying frequencies and directions. In this work, we represent initial perturbations as a set of independent wave modes, each evolving according to the system dynamics. This decomposition is particularly effective in the linear regime, where small-amplitude assumptions allow the neglect of mode interactions and enable independent analysis of each component [40, 56]. By adopting plane wave perspective, we obtain tractable and insightful view for understanding perturbation evolution in the Ulasov-Poisson system.
Contribution. In this paper, we introduce a novel solution to the OOD problems by harnessing collective behavior dynamics modeled through the Vlasov-Poisson system. Departing from conven- tional approaches, our method conducts a theoretical analysis of the basin of attraction's 's boundary for steady state solutions, elucidating its effectiveness in addressing OOD detection task. Additionally, examine perturbation evolution within dynamical systems and propose strategy to expand the
18

---

basin of attraction, thereby tackling OOD generalization challenges. Extensive empirical experiments validate the efficacy of our approach.
B Proof
B.1 Formal Proof of Theorem 2.1
Consider the Vlasov-Poisson system Z-dimensional space:
(.(z.v.t) +V VzF(z, v,t) E(z,t) v.F(z,v,t) = 0,
(20)
δt
V))(z1)) FF( dv
E(z,t) = Vo(2,))
(21)
where:
z, V E TRZ are the spatial and velocity coordinates, respectively.
F(z.v, v.t) R ? R R is the particle distribution function.
E(z,t) E R is the electric field.
o(z,t) t) Rd x R+ + R is the electric potential.
co > 0 the permittivity constant.
The particle density is:
p(z,t) t) F(z.v,t)dv.
(22)
We study N systems indexed by = 1,.... NN, each with:
Particles of identical mass (normalized to 1) and velocity vo IR..
Spatial density n2(z) E L' (R?) n L (R² with nı(z) & nz(z) for i
Distribution (z v,t) (z(66((v vo). where ö(v vo) is the Dirac delta distribution
in R
Ion background density i = nz(
steady state solution satisfies aF, distinct for distinct nz) Assumption B.I. For each system
0. We aim to prove that the steady state solutions Fj are
0t
1. Paun, (z) = nı(2)) ensuring E = O. 2. Vo (z) ensuring F; steady state.
Proof. Let consider two systems and j with steady state distributions:
(z,z, v) n. (z)) vo), Fz ny (z) vo).
(23)
We need to show that n. (z) + (z), then F, * Fj in the sense of distributions.
First, verify that F; and F, are steady state solutions under the given assumptions. With Pion, (z) nz(2), compute the particle density:
=
(z = / Fl(z,))do = / /n((2)0VV - -ooiiivvoooooooo = dv nz(z).
(24)
The Poisson equation becomes:
pz(z) (1,(2)
(2)
(25)
eo
60
Since Pzon,i(z) = (z),,, the total charge density is:
Protal, (z) Pion. (z) pz(z) =2,(2) (z) = 0.
(26)
19

---

Thus:
(z)(zz) = Piotal,(Zz) 0.
(27)
60
Assuming boundary conditions such that Φ; - as |z| oc, the solution is (.(z) 0, so:
E (z) V (z( 0.
(28)
Similarly, Ej = for system j.
With E, = 0, the Vlasov equation for system i reduces to:
oR:
V2 0.
(29)
0t
For steady state,
aF tt
0, requiring:
V2F; = O.
(30)
Compute the spatial gradient:
V.P. Velni(z)0(v vo)] (V2ni(2))0(v -vo),
(31)
SO:
v V.F. vV V2n((z)o(( vo).
(32)
Evaluate Vo:
Vo (z)
(33)
which holds by assumption. Thus, v V.F, 0, confirming that F; is steady state solution. The same applies to F..
Now, address distinctness. Since n2(z)) # (z) there exists set S C R of positive Lebesgue
measure where (z) * (z).). The velocity component 8(v vo) is identical across systems.
Consider the action of F; and Fj as distributions on test function (z(z, v) C (IR R)::
(Fi, = 1000))//aaa nz(z)ō(v vo)u v) dz [ n.(z) vo) dz.
(34)
Similarly:
(F), 1 nz(z)a(z, vo) dz.
(35)
If F. Fy. then (F,, (Fzs et) for all vi implying:
f [n;(z) (n vo) dz = 0.
(36)
Choose (2 v) = (v vo), where x(z) smooth function supported in S and 6, is mollified delta function. As + 0, this forces n. (z) = (z)(z) on S, contradicting the assumption that n4 A n, on a set positive measure. Thus, F, + Fj as distributions.
This theory posits that high-level features from different distributions correspond to distinct steady state solutions, meaning that OOD features and InD features cannot share the same steady state solution, thereby making it particularly applicable to OOD detection tasks.
Remark.
B.2 Formal Proof of Theorem 2.2
We derive the nonlinear
equation for the steady state Vlasov- Poisson system
in thermodynamic equilibrium, where the background ion density (z) R + R+ varies spatially. In thermodynamic equilibrium, the steady state distribution function F Fz v) takes the Maxwell-Boltzman form, reflecting the energy distribution of particles at constant temperature.
Assume the steady state distribution is:
F*(2,v) = Aexp (-1 mm1ll (-1
(0)))
(37)
where a normalization constant, m > 0 the particle mass, and di (z) is the steady-state electric potential. The factor [mll2v| represents the kinetic energy, and o (z) is scaled such that
20

---

the potential energy term gΦ (kT) x (z), with q as the particle charge, h as the Boltzmann
constant, and T as the temperature.
Compute the particle density by integrating over velocity space:
p(z) 1 =e
= Ae dx)
(x) J exp
(-Eml(vl?) dv.
(38)
The velocity integral is Z-dimensional Gaussian. For each dimension ..... Z, evaluate:
exp (-2mo?? dv;
2m
(39)
m
since S_o e=as" dr = V a/a with a = mm/2. Thus:
1 exp 1 m-1m) dv dv= (v
2
7/2
2m
2z
(40)
m
m
The particle density becomes:
2x
7/2
p(z) = A
e xx (x) = Ce (x)
(41)
where C
A (1 [xx] 2/2
is positive constant. To absorb C into the scaling of the charge density,
m
effectively setting C' = 1 in the units used, so it becomes:
p(z) = e (2)
(42)
The steady state Poisson equation is:
1 (p(z) Pion (Z)n(z))
V²O'(z) = (z)=
(43)
60
Substituting p(z), we obtain:
(z)
(e- Pion -pto(()))
(44)
6o
This is the nonlinear and the charge densities in the presence of spatially varying ion background.
equation, capturing the relationship between the potential
To confirm that F* (z, v) is a steady state solution, verify the Vlasov equation with OF
0:
v V. (z, v) + E(z) (z, O.
(45)
Compute the spatial gradient:
F2 = FF, V
(46)
S0:
(2 -F (z))
(47)
For the velocity gradient:
V.F v) -m,)"(a,v),
(48)
and with E(z) = - (z):
E(z) Fz, = (-m())))-mmm( = mF (v-x(((22
(49)
Summing the terms:
-F v) (v V (z)) + mFF* (2, v) (v- vZ) (m-1))F (v. (z) (50)
In the scaled units where the potential accounts for q/ (kT)), the coefficients balance (effectively setting m 1 in the normalized system), yielding zero, confirming the steady-state condition.
Thus, the nonlinear Poisson-Boltzmann equation is:
V (2) - (e (0-)(()) Pion (hom(z))
(51)
cD
completing the proof.
21

---

B.3 Formal Proof of Corollary 2.3
We establish the properties of Lyapunov functional for the
system in thermodynamic equilibrium and characterize the boundary of the basin of attraction for the steady state (Ø", F*), where (z, v) A exp satisfying the
2 m |Ivl o² 0²(z) and the particle density is scaled as p(z) = e o' (x)
equation (²) - (e-0"
(z) phon(z))
co
Define the total free-energy functional for general state (ő(2, F();
Cu /
1 /
(1))
|/2 |V ()] dz
J3z F(z, v,
+ ((2, +
dv dz.
(52)
To confirm (t) Lyapunov functional, compute its time derivative. Using the Vlasov equation aF de
+
V F - (Vø) V.F = 0, the second term's 's derivative is:
d
1 F
m + + dv dz = J JoL
OF (1 (m +0) + FF as
dv dz.
dt
RZ
JR2
0t
01
(53)
Substitute
aF at"
The term with m||lv| becomes: becomess:
J&e (-v V2F V+F): 2
1 =mllvvl [ll dz.
(vv
(54)
Integration by parts, assuming F - 0 |z). |v| + oc. nullifies the Vz term, and since V. (2 [v|?)] mv, the velocity term is:
m / Vo.
(/e Fivrr dz.
(55)
R22
The @ term contributes:
Jez J (-v V-v.V+++(VV V.Fl dz +
/
do P
dz,
(56)
ai
R2
with the Vv term vamishing. For the electrostatic term, use the Poisson equation:
d 6n |V dz)
| (p- -ion dz.
Oo
(57)
dt
J22
Øt
Combining, dissipative terms ensure di non-increasing and minimal at St For the steady state, compute L using Fr. The velocity integral is:
d (t) < 0, with equality at (o, F) (D), making 2t())
Jaz
F
1 + 0²)) dv = e (x)
2 m
2 m
2m
m
2/2
2
2
65 (z)
1TT 2/2
m
(58)
Thus:
L
+ (e-d(()) [z)
Plon
55 dz.
(59)
R2 2
Define the energy density:
E(z)
60 2
(z)(2) + (6- - Prna()) -Pro0 0= (z)),
(60)
so -== = S2z Elz) dz. Zet be where x= (z) is minimal, with 17 E(z") ). The energy defect is:
g( (z)" E(2) 1 (z)(2)) (c (c-0 Prva (z) o² (z) 17-
(61)
Since lt) is non-increasing, g is non-negative and decreases along non-stationary phase-space characteristics (z(t), v(t). At z", V (z") = 0, so G O.
For an initial point zo. if G( (2o)) < O, the trajectory remains in the sub-level set containing z converging to the steady state, so 2n E B. If G(Ø" (zo)) > 0, the trajectory cannot cross G = 0, staying outside B. Thus, the basin boundary is:
dB {z R 9 (z)) 0}.
(62)
22

---

Since 6- is smooth (by elliptic regularity), G is differentiable. Near åB, the gradient V=g(Ø"(z) (z)) changes sharply, as the rapid transition from G > 0 to G < 0 implies large magnitude of V&G = _ Eo Vi + (e Piea) - x e o1 driven by the nonlinear terms and the Pois- son-Boltzmann equation. This sharp gradient reflects the boundary's 's role as a critical separator, making öB differentiable level set, completing the proof.
B.4 Formal Proof of Theorem 2.4
Prior to proving the Theorem 2.4, we introduce two prerequisite lemmas:
Lemma B.A (Sobolev Embedding Theorem). Let Q C R2 be a bounded domain with a smooth boundary. For a function u E ( where (Qt) is the Sobolev space of order s, the following embeddings hold:
If Z/2, then (Q)) the space of continuous functions Ω2
Ifs> > Z/2+ k. for positive integer then u E ( the space of k-times continuously differentiable functions on Q
Proof Lemma B.A. We prove the Sobolev embedding theorem for a bounded domain C RR2 with a smooth boundary. focusing on the cases relevant to our application: continuity (s > 7/2) and C differentiability (s > Z/2 + 2). The Sobolev space H H (Ω) consists of functions E (22
whose weak derivatives up to order are in (S2), with the norm defined via the Fourier transform of an extended function. Since Q is bounded, we extend u E H to ū E H (IR with compact support, using standard extension operator that preserves the Sobolev norm up constant.
The Fourier transform of ū is ū(k) = SAz ū(Z)E d dz. The (R?? norm is:
/ 1 (1 |k| ((k) dk < 80.
(63)
To show continuity for 8 > 2/2, express iiz) via the inverse Fourier transform:
ū(z) = (2x) 2/2
JRZ (kk) dk.
(64)
We need to prove is continuous, i.e.. |ū(21) - ū((z)) - 0 as 21
- Z2. Consider:
J ((
(z))) ((2)2 (2mr 2/2
x dk
(65)
Since le-k-z
kk 22 < 2, and for small |z| 22/, we use le eik-zo| S |k| [z]
12/. split the
integral into low- www-frequecy (|]k|| s R) and high-ffreuency (|| > R) parts:
(NAEER [ū(k)] -2 dk + 2 / /
((2)) - ū(z2)) S (2/)-2/2
2/2
(
[ū(k)/
(66)
For the low-frequency part, apply Cauchy-Schwarr
(TINISS
1/2
(TINISS
||k|12 +||1222) dk
1/2
(())||k(H dk
((k)) + [||(/) dk
(1
[kig
(67)
The first integral bounded by || The second is finite if 2s - Z, i.e., 8 >> 2/2, 2, as the integrand behaves like || for large ||k|] For the high-frequenee part:
( MAISR
1/2
(
1/2
((k) dk S
(((k) |2 dk
VIRIDRR
(1+ [|| dk
|
(68)
The second integral converges for > 2/2, and the first is controlled by ||]]| As R - the high-frequency term vanishes, and the low-frequency term is proportional to |z] - 22/. proving continuity.
23

---

For C differentiabilityy. if >> Z/2 + k, consider the weak derivative D for |a] S k. Its Fourier transform is (ik)) and:
| (1+
||al||| {()l|((( dk - / (1 + |k (k)))) dk,
(69)
R
since |k| I (1 + 3 |l| If s |a| > Z/2, then D E Ha C CO so ū E (R For |a|| ki, we need k > 2/2, i.e., > 2/2 - k. Restricting to Q, the extension operator ensures E C* (22), completing the proof.
Lemma B.B (Measure of Sublevel Sets). Let Z C. I2 be bounded open set with smooth boundary, and let f.g E (St) s > Z/2, with f E (2), E 0 (St), and 81 > so. For a fixed threshold 5 > 0, define the sublevel sets:
S {z E [1) < 6), S = fz EQ [9(2)) < 63.
(70)
If g has non-trivial higu-freeuency Fourier components, then the Lebesgue measure satisfies: meas(S)) Z meas((,),
(71)
with strict inequality if high-frequency components of g cause [g] to exceed 6 in regions where |/| <
Proof of Lemma B.B. We prove that the Lebesgue measure of the sublevel S {z Q
[z(z] < 6} for a function f E H (22) is at least as large as that of Sg = {z E Ω [g()) < 5} for g E (St),, where 81 > 8p > 7/2, and Ω C TR² is bounded with a smooth boundary. Since 50 2/2, the Sobolev embedding theorem (cf. Lemma B.A) ensures f.g E C)2)), so the sublevel
sets are well-defined, and their Lebesgue measures are finite as meas (9t) < co.
The smoothness of and is characterized by their Sobolev norms:
11 = [ (1 + |||] ((kk) dk.
Ilglirro /
(1+ (k)))))0)((k)) dk.
(72)
where 9. are the Fourier transforms of the extensions of f.g R2 via a bounded extension operator. Since 81 Su, the decay of [f(kk) for large |k|| is faster than that of lā(k),, implying f has fewer high-frequency oscillations. High-frequency components in y can cause rapid fluctuations, potentially pushing [z) 2 d in regions where [s() < 6.
Consider the sublevel sets. The complement (S E [f(z)) 2 8} has measure:
meas( \S))))
dz.
(73)
(2E9115())22)
By Chebyshev's inequality for [f]: meas((z (z E e [l11((z)] 81) 8 (f(2)
8z I1
(74)
Since 8h: C T.2, and similarly for g. the 1,2-norm bounds the measure of the complement. However, smoothness affects the distribution of l.f]. For g E H high-frequeng, modes (larger []kk for large ||k|] can create localized oscillations, increasing the set where [g(z) 2 8. For H faster Fourier decay reduces such oscillations, shrinking Q Sy. To quantify, assume has non-trivial high-frequency components, i.e., J | (1+ ||k| [(k) dk > 0 for large R. These components contribute fluctuations in g. modeled as g Slow + Shigh. where Jhigh corresponds to |k|| > R. In regions where !9tow| < 8, oscillations in 9high may push [gl 2 6, reducing mocas(Sg). For f. with $1 So. the higu-frey contribution is smaller, mcas( Sy) S mcas(s? (S2), hence:
meas((S)) = me) - meas(S) \ (S)) 2 meas(?) meas( \ Syy) = meas(S,).
(75)
Strict incquality holds if Shigh causes [gl 2 in a set of positive measure where Ifl < ô, completing the proof.
Proof of Theorem 2.4. We prove that increasing the spectral regularization strength Adisp > 0 (in other words, 3 > 0) in the dispersion relation loss Laisp results in strictly larger Lebesgue measure
for the approximate attractor basin Bd {z Ω (V² )z()) + ep - ^}, compared to the
24

---

case with Adisp = O. The proof relies on the smoothness induced by Cuisp on the potential og, which reduces oscillations in the residual field, thereby expanding the sublevel set.
Consider the steady state Vlasov-Poisson system in thermodynamic equilibrium, where the distribu- tion function is Fo v) A exp (- mm]l/] (z))) and the particle density is scaled as p(z)
e e The steady state Poisoon-Boltzmann equation is VZ
do(2) pion pron(z)) (z)). The
approximate potential @o is trained to minimize losses, including:
Prsisson
(2; 2; for 00) V200(z) e (
(76)
c0
and the dispersion relation loss:
Ldisp 2: N0) = Σ || +
(77)
Kx(2) (z ikz (=)===
= e )o(x)) k
The term Ldisp penalizes deviations from the equilibrium dispersion relation in Fourier space, with the |k||² factor emphasizing high-frequency modes. Minimizing Laisp enforces rapid decay of the Fourier coefficients of og. For fixed Px. consider the Fourier transform Ok Sa (0 dz. Minimizing Σx |0x// (from the ||<|//% term) implies:
E 1K13)))))) oc, k
(78)
for some O, placing dg (S2),, the Sobolev space of order With Adisp 0, the penalty high-frequency modes increases s, enhancing the smoothness of dy compared to Adisp = 0, where only Lpoisson and the Vlasov loss Lvlasov active.
By the Sobolev embedding theorem (cf. Lemma B.A), for 5 C R2Z if Z/2, then de H is continuous, and if s > Z/2 2, then @o E (?)), ensuring the Laplacian V2000 and the nonlinear term e θ0 are well-defined and continuous. Increased smoothness reduces oscillations in de and its derivatives. Define the residual field:
R(z) V2
_ (0( 60
(79)
Since R(z) depends & and d0 both of which inherit the smoothness of de. higher (from Adiop > 0) makes R(zz) smoother, with fewer sharp spikes or oscillations.
The approximate basin is:
B₁ {z E 12 <
- (2 < VXJ.
(80)
In measure theory (cf. Lemma B.B), the Lebesgue measure of sublevel set {z E Ω [f(z)) < 8} increases as the function f becomes smoother, because smoothness reduces regions where ff exceeds ó due local oscillations. For R(z). high-frequency oscillations in the or V2 can cause [[(x] to spike above VÀ, shrinking Bg. With Adisp > the enhanced smoothness of de (higher s) reduces such spikes, as the Fourier modes decay faster, leading to more uniform R(z).
To formalize, consider two potentials: %0 trained with Adisp > 0, and o with Aaisp 0. Let (2) and (2) be their residuals, with ob E H (2), (0) E (S)),, and 81 > 80- The smoother (2()) has larger sublevel set {z [Z)( < VĀJ because its fluctuations are less likely
exceed VX. Thus:
mcas (B()an200) M mcas (B((a)=)))
(81)
To ensure strict inequality, assume dg has non-trivial high-frequency components (common in numerical approximations without regularization), causing ()) to exceed v in regions where R()) remains below due to damping. This yields a strictly larger measure for Bo(Am>>0) completing the proof.
25

---

B.5
Formal Proof of Theorem 2.5
Prior to presenting the proof, we introduce the following lemma:
Lemma B.C (Standard Statistical Learning Theory for Error Bound), Let X C IR be a bounded domain with a smooth boundary. Let hgg + IR.,, parameterized by 0, be a model in (X),), > d/2, approximating true function he E (X). Let x ~ D be i.i.d training data from X. For a bounded. Lipschitz continuous loss function { he). the generalization emror:
Egen(0) (0) = EEnD [l]ba (2) - 1²" (x)11
(82)
is bounded by:
(0) S Eraai (0) Cilholl
(83)
where Emainn EEwD [e(x;; he]] is the training error, and C 0 depends on the loss Lipschitz
constant and model class complexity.
Proof of Lemma B.C. We bound the generalization error for model he E H*(X), s d/2, approximating i E (X). The generalization error is:
Egen(( (0) = Ex [1]b(()) - (x))1
(84)
The training error is:
E (0) Ex (e(x; ho)]
(85)
In statistical learning theory, the generalization error is bounded by the training error plus term reflecting the function class complexity. Since & > d/2, the Sobolev embedding (similar to Theorem 2.4) ensures he, ht C*(X). the squared error well-defined. Assume E(r; ha) is Lipschitz continuous with constant L, i.e., [e(x; h1) - {(; h2)l s L|]h1 h2ll. and bounded, ||| s M.
For i..d. samples 24, the empirical risk is:
n Σ ho). 12
Exain (0)
(86)
By uniform convergence (e.g., via Rademacher complexity). for the function class H {ho 0 E +]:
|Eram (0) Eurain (f)) S (H),
(87)
where R,,(H) is the Rademacher complexity. For H (X), the Sobolev norm bounds the magnitude:
|Aall S Chllhll
(88)
Thus:
Ra S Ca sup [Ihal| hoEH
(89)
Assuming ((x; ho) = [1))(x) - h0t h²t (2) or bounds it, the generalization satisfies:
(g) S Eurin (0) + Callhollas,
(90)
where C4 depends I., D, and sup || ||
m². Thus:
Egen (0) S Eirain + Cl
(91)
completing the proof.
Proof of Theorem 2.5. We prove that including the dispersion relation loss Ldisp reduces the generalization eтror (0) by constraining the - of de, leveraging the "lasov-Poisson
system in thermodynamic equilibrium where Fo v) A cxp (- 1 |||lv|| @o(2)) p(z) e>bo(z) and the true solutions Ø², F* satisfy the Doisson-Boltznen
equation.
The generalization error measures the expected squared deviation of the learned solutions de Fg from the true solutions:
(g)n( E, [100(z) d" (2)1 + Ez, [|o62, v) - (2,
(92)
By standard statistical learning theory (cf. Lemma B.C), the generalization error is bounded by:
Egen (0) Eurain (0) + Complexity(de,.F))
(93)
26

---

where ((aan(() = Ez,v [Croisson 0o) + X1 Ciaso (z: fo: 0g. Fo) + Adisp Caspp (z; Øo)] and the complexity term depends on the Sobolev norms || l Fa Small training errors imply @g R 6², F' F*, but the complexity term, which measures model capacity, must be controlled to prevent overfitting.
Expanding the squared norm of Ldiup (z: far 0a):
kll oe (z)e i-z + e k -
= [k])" + 22o(()) + | 2 (z)e
(94)
Take the expectation over z ~ D and k with probability density p(k) (e.g., a Gaussian distribution over wavevectors):
(uunpp
Σ [B [[]]]*] + E2 [e 200(x) + 2E (z)( vo(2) Ex [[]k]²]
(95)
Since Z/2, Lemma B.A ensures Φe E (S2), so (0( and ov(() are bounded. The cross
term is bounded by:
Ez (b(())e (z))) (x) Ek [||k| - Ci.
(96)
where 1 = | I| l|e de = Ex [|]]||]] The term Ez [e 2x) 2 O. For the first term, consider
the Fourier representation:
Ex [8o(2]) (2) Ex [||]||]] = J (0)(2)) 2 dz 1E [[]]]]]] ca llbb (0)
(97)
where C₁ LEk [|] > O. In the Fourier domain, the term involving |k|| is:
{ |k (k))) dk 1| 60 1/.2.
Σ |k|| (k)()2
(98)
Thus:
Bx Lddipp 2 ci 1100 22(0)) + Ey 200(2) -
(99)
Since |100 approximate:
10
| + 2| l17 and the dominant term is || 2o
11,,2 we
Eg,k disp 2 cllooll -
(100)
where c = C1, and C bounds the nonlinear and cross terms, since Ez [c-260] 2 and C1 is finite.
The Vlasov loss LViisov constrains || VVVollze. V/ and Fellza, but not directly || or [0øl|lz.. Thus, without Edisp- [| Ha may be large due to high- figeffrrguency components. Minimizing Edisp bounds:
Ez., E (Lissp] + C.
(101)
For the generalization error, consider:
Eg [l0o(z) o" (z) S E8 [Croisson] C2lloollx*
(102)
since LPoisson measures the residual, and the Sobolev norm controls model complexity. Similarly, Vilasey ensures Fg Rs F", with complexity bounded by (F. Thus:
Egen
Eraain (0) + C's (lléo ( [|Foll
(103)
Since Laisp reduces || H² via its frequency-doman penalty, and He C H2 for > 2, it lowers the complexity term, reducing Egen (0), completing the proof.
B.6 Formal Proof of Corollary 2.6
Before proceeding with the proof, we first introduce two lemmas. Lemma B.D (MeDiarmid's Inequality). ) Let X C Rd be a bounded domain, and let I1, In D be i.i.d. samples from X. Let f X + R. be a function such that for all changing one sample I; to x alters f by at most i.e., [f ...
Zi. any 8 (0, 1), with probability least 8:
In) ) - f(z1..... ,",..., S c₁. Then, for
> Di=
O log (2/))
Elf ., xw,1 f(r (2
(104)
2
27

---

Proof of Lemma B.D. Let f X + R satisfy the bounded difference condition: for each [f(z)...., ................................ Fi...,,
G. Define the random variable Z (111..., Rn), where ~ D. Z is concentrated around its expectation. The variance proxy bounded by the sum of squared differences: Σi=1 et. Applying McDiarmid's inequality, for any
t >> 0:
2t
P(Z - E1 2 1) S exp
(105)
n
Set the right-hand side to 6/2:
21
8
exp
(106)
i=1
Solving:
- & log(2/6)
(107)
Similarly, P( - Z 2 1) S 6/2. By the union bound, with probability at least 1 - 6:
(2/()
E E 2
(108)
2
Thus:
Σ Li=t &f log log(2/0)
(109)
completing the proof.
Lemma B.E (Rademacher Complexity Bound), Let X R.d be bounded domain, and let T1,.... ~ 'D be i.i.d. samples from X. Let H {ho -X } be a hypothesis class in (X), s > d/2, and let E: IR" x + R be a loss function bounded by M and Lipschitz continuous with constant L. The generalization error Eg [e(ho, x)) for he E H. is bounded, with probability at least 8, by:
}
+o(1 (og()/))
(V
Eg x) S Σ (ho, xz) + 2Rad(Cx) +0
(110)
where Rad( Rad(Cu) E,, (supheen Σn of((ho, is the Rademacher complexity of the loss class Cx {x E(he, x) he H}.
Proof of Lemma B.E. For H C H*(X). 8 >> d/2, the loss E(ha, x) bounded, |&] S M, and Lipschitz continuous, (e(ho, ((ho,,3) 5:] S To (x) ho, (1)]l. The empirical risk is:
ε(ho)
((ba xz).
(111)
The expected risk is E(he) E2 (e x) By standard results in statistical learning, the generaliza- tion gap is bounded using Rademacher complexity:
Rad(Cn) = Egg
(112)
hoEH
i=1
where o; Unif((-1,1)). For any ho, with probability at least 1 5:
12g(2/0)
E(ha) E(ha) 2Rad(Lx) + C.V
(113)
n
using Talagrand's lemma and concentration. Since is bounded, apply Lemma B.D to the function f 2n)) suph, E ((ea, I.), with 2M The additional term is:
G Σn n2 log(2/8) -Ci
4M2 log(2/5)
zo (V [log(1/5)
(114)
2
2n
n2
28

---

Thus:
E(ho) - E(ho) + (Rad(Cx) + (v (10108(1/0)
(115)
completing the proof.
Proof of Corollary 2.6. We derive the generalization bound for the Vlasov-Poisson system under the assumptions of Theorem 2.5, where Φg, Feg [I (R? x IR ). s > Z/2, approximate the true solutions Ø² F", and training data (z, vi) ~ DD are i.i.d. from ft x IR The hypothesis class is H {(0.. F9) E )), and the generalization error (q.)) We aim bound (d) and show that Ldisp reduces the Rademacher complexity Rad(H().
Define the loss function for hypothesis ha (00, Fa) H::
((o,, (z, v)) (d)(z) (z)) Fo v) (z.
(116)
The generalization error is the expected loss:
Egen (0) Ex,v [e(hg, (z, v))l
(117)
The empirical risk over n samples is:
gen (0)
Σ 1=1
E(ha. (7,, ve)).
(118)
By statistical learning theory, we bound the difference between expected and empirical risk. The Rademacher complexity of the loss class Li =1(z, v) + E(ho, (z, v)) ho E H) is:
Rad(Lx) = E.RRY
sup NEEH
Σ
ar (zi, vi))
(119)
where o; ~ Unif((-1113) are Rademacher variables. Since is the squared error, assume it is bounded (as 7/2 ensures (bg; Fe E C by Theorem 2.4, and Ω is bounded). The loss is Lipschitz continuous:
|e(ho) (z, v)) E(hez, (z, v))1 S (los, (z) 60l FFo (z, v) Fe, (z, v))),
(120)
where L = 2 max 60 o² 1=>,,, Fg
F =]
Using standard results (cf. Lemma B.D and Lemma B.E), with probability at least 1 8:
Egen(0) < Egen( (0) + 2Rad( + + log(1/5)
(121)
where C1 depends on the bound of f. Since Cpoisson and LVlasov approximate the squared errors (small residuals imply dh 0² Fe se F*), and Ldisp adds regularization, the training loss L bounds é. Thus, the empirical training error Euraia (0) E. L(z,, Vs: dg. Fo) satisfies:
Egen( < &nain (0) + C2,
(122)
where C2 accounts for differences between and L. Taking expectations:
E gen (0) S Eirain (0) + 2Rad(Cx) + Ci log(1/5) + Cz.
(123)
Since Rad CH) depends on H. we denote Rad(H) Rad(Lx), and absorb C2 into the O(/log(1/8)/n) term, yielding:
(g) Evain (0) 2Rad(H) +0 (V log(1/6)
(124)
Now, we show that Ldisp reduces (ad(H). The Rademacher complexity bounded by the complexity of H:
Rad(H) S C3 sup ||00| + sup ||Fo //V.
(125)
dbu
29

---

By Theorem 2.5, dissp satisfies:
Kx, Edisp] 2 cllobolis C.
(126)
Minimizing Laisp constrains:
110lli12 I _Bzk(Cdisp! + +
(127)
Define the subset Hs = (0o E H² Ez, Liting) - M) for some bound M. Then:
M C+C
1 S
(128)
restricting Og to a bounded subset of H Since H C. H² for s 2 2, this reduces sup 1 |-. For Fg = A exp 1-2mll/v] Φ0(z)), the norm Folla depends on [ constraining Φe also bounds Fg. Thus, Ldisp reduces Rad(H), tightening the generalization bound, completing the proof.
C Algorithm Protocol
Algo. 1 and Algo. give the algorithmic protocol our framework, which is easy implement and applicable to common OOD problems.
Algorithm Algorithm pseudocode of CBD-De
Require: Training mini-batches (v:, yi ] pre-trained feature extractor for distribution prediction head Fe. potential prediction head Φe. learning rate 1)
Ensure: Trained parameters
1: Initialize 0 <- (Fo; ) optimizer O(0,n) /* Fixed fo, post-hoc fashion +/
2: for each training step do
3: x <- {xx))=>> y Y {
4:
+ fe(x) /* Extract features */
5:
v - ones _ike(z) /* Constant velocity input */
Vlasov residual: v. V. - Vo F
6:
F Vo(z.v)
7:
O* + (0
8:
V Ve0* + autograd (itograd((".summ)....
9:
E. + V2d"
10:
VxF*, V.F* autograd F (r..ammFFsmmmmmm (z. vl)
11:
V + mean (v V,F* + E Vr.F)
Poisson residual:
Vi + exp(-o")
12:
VeØ" t- autograd sum() 2)
13:
/* Compute second-order derivatives each dim */
14:
for d= 1 do
15:
V + 06" /Ōza + autograd sum(), za)
16:
00 102
(Vo) zd)
17:
Accumulate 22 + V2 + + 10000/020 2
18:
end for
19:
Cpoisoon + mean @eann(VV20 cxp(-")))
Total loss and backpropagation
20:
Liotal f- Lylasor + Lpoison
21:
backwaac O.step()
22:
23:
O.zero_grad()
24: end for
30

---

Algorithm 2 Algorithm pseudocode of CBD-Gen
Require:
Training mini-batches { } feature extractor for classifier head Sg. distribution
prediction head Fe, potential prediction head dg, learning rate 7,, loss weight a, plane waves number N, scaling factor o
Ensure: Trained parameters 0
1: Initialize 0 {fo Sa, Fo, Φ0), optimizer ((0,,) n))
2: for each training step do
3: +- {m y - [=
4:
fo(x) / Extract features
5:
v - ones like(z) /* Constant velocity input */
Vlasov residual:
v. v.V.F Vp F
6:
F Fo(
7:
O" + (z
8:
V VeΦ - autograd summ)
9:
E. + Ve0
VxF", V.F* aFtograd(
2. v|)
11:
fa mean (v VaF* + E V. F*)
Poisson residual: Vi + exp(-o")
12:
V V2Ø"= t- autograd (p" sum(),
Compute second-order derivatives on each dim
14:
for d d to Z do
V + 0 /0z/ + autograd O- summ), za)
00 /0= autograd (Vo summ zd)
17:
Accumulate V V2 + 0 0000 1
18:
end for Cpoisoon mean ( exp(-
19:
Dispersion relation loss:
Σx / + pill
Sample [ NN IN
21:
for each k; do
σ. < exp(p=p [jj k. z)
23:
6%. mean (b" a;),
Pk; mean(exp(-d)) .0))
24:
F: |1 0F: Pk,
25:
end for
26:
Ldisp sum sum((r;
Classification loss
27:
ypred + (z) Celnsiity CrossEntropy(yyee.
28:
Total loss and backpropagation
29:
Liotal < Laussity + ( ppissonn) + BL
30:
(ackwwack)ard))
31:
O.step()
32:
O.zero_grad()
33: end for
31

---

D Additional Experiment Settings and Details
We conduct experiments following the latest version of OpenOOD' [96, 94] and the DomainBed [23]. In this section, we first provide more details for the utilized baselines (Section D.1), datasets and evaluation protocol (Section D.2), and model architectures (Section D.3). Then, we demonstrate the hyperparameters (Section D.4) and thw experiment infrastructures (Section D.5). Finally, we include additional ablation and validation results to better understand the proposed method, including the effect of the initial velocity assumption (Section D.6), sensitivity analysis of the MLP architecture (Section D.7), and the validation of the MLP steady-state approximation (Section D.8).
D.1 Baselines
OOD Detection. To evaluate the effectiveness of CBD-De in detecting OOD inputs, we conduct extensive comparisons with 22 representative post-hoc OOD detection methods across three widely adopted benchmarks: CIFAR-10, 10, CIFAR-100) 100, and ImageNet- 1k. These baselines span a range of methodological categories, including confidence-bas based methods such as OpenMax [6] and MSP [28]; calibration and scaling techniques like TempScale [24] and ODIN [47]; distance- based and density-
aware approaches such as MDS [43], MDSEns [43], RMDS [62], and KNN [77]: feature-based
scoring methods including Gram [66]. ReAct [75]. RankFeat [73]. and VIM [85]; energy- and
activation-basee models such as EBO [50], MLS [26]. KLM [26], and ASH [14]; generative and
hybrid models like OpenGAN [37]. SHE [97]. and GEN [5 as well as recent advances in neuron-
level uncertainty estimation such as GradNorm [32], DICE [76], and NAC-UE [52]. All evaluation results reported in Tables 13, 14, and 15 are derived from the standardized OpenOOD implementations to ensure fair and reproducible comparisons.
OOD Generalization. We evaluate the OOD generalization capability of CBD-Gen by comparing
with 17 representative baselines spanning diverse algorithmic categories, including distribution matching (MMD [46]), distributionally robust optimization (GroupDRO [65], VREx [39], IRM [2]), data augmentation (MixStyle [103], ARM [98], Mixup [91].. representation learning (SagNet [58], MTL [7]. MLDG [44]), alignment-based methods (CORAL [74]), empirical risk minimization (ERM [80], feature suppression and invariance strategies (RSC [34]), as well as sharpness-aware optimization techniques (SAM [68]. GSAM [104], SAGM [86]) and gradient-guides approaches (GGA [3). Comprehensive experimental results across five widely used OOD generalization bench-
marks (PACS, VLCS, OfficeHome, Terralnc, and DomainNet) summarized in Tables 16 to 20.
Comparison to Traditional Statistical Methods. Beyond empirical performance, it is essential to contextualize our proposed method within the broader landscape of traditional statistical approaches for OOD learning. While most conventional methods (e.g., energy-based modeling, density estimation, or confidence scoring) focus on point-wise statistics or local sample relationships, they typically overlook the structured interactions among features or samples. In contrast, our Vlasov-Poisson formulation provides principled, physically interpretable mechanism that inherently captures global inter-feature dynamics through self-consistent field evolution. This collective interaction allows our model to reflect structural dependencies that classical cooring-based probabilistic models cannot explicitly encode. Table 5 summarizes this conceptual distinction. We compare our method with several representative approaches across multiple dimensions, including their modeling assumptions, underlying OOD mechanisms, sample interaction properties, and interpretability.
D.2 Benchmarks
OOD Detection. We primarily evaluate CBD-De using the Far-OOD track from the OpenOOD benchmark suite, which offers standardized and well-established evaluation protocol adopted by numerous prior works.
CIFAR-10 and CIF,R-100, consisting of 10 and 100 classes respectively, serve as the in-distribution (InD) datasets in our experiments. Following the official OpenOOD setup, we adopt a consistent data split for both benchmarks. Specifically, we use the full training set (50,000 images) for model training. From the official test set, 1,000 samples are reserved as the InD validation set, while the remaining 9,000 images are used as the InD test set. For OOD validation, we follow the OpenOOD protocol by
'https:// ://i thub. www/Ji www.///gganggg0////oooo Openoon. 'nttps: http:////...coooteeeeeeee/eaaaaa
32

---

Table 5: Comparison between our CBD- method and traditional statistical OOD Approaches.
Approaches
Modeling Assamp-
OOD
Mecha-
Sample
Interac-
Key Formula
Interpretahility
tion
nism
tions
Eieegy-hused
Energy functions
Low energy -
[X] No interac-
{(x) (e(r
Intuitive locally, hard globally
Models
feature space
InD, high 7
tions
OOD
Density Estima- tion
Probability sity modeling
den-
Low
density
[A] Local interae- tions via density
p(.)) I(,(()))
Captures local treads. lacks high- dim clarity
- OOD
Confidence Scor- Model output con-
Low
confi-
[X] No interac- tions
C(a) softmax(
Heuristic, limited by generalization
ing
fidence
dence + OOD
Ours
Particles self-
Cannot
reach
[-] Gilobal
ex-
3r VVVVEE
Physics- via particle dynamics
Polsoooo
consistent field
steady state
plicit interactions
selecting 1,000 images from Tiny ImageNet [41], evenly sampled across 20 unseen categories. This ensures a controlled setting for hyperparameter tuning and model calibration. To comprehensively assess the generalization ability of OOD detection methods, we evaluate performance on four widely used OOD test datasets each disjoint from both the InD and OOD validation sets. These datasets include:
MNIST [13] is a benchmark dataset for handwritten digit classification, consisting of 70,000 grayscale images of digits (60,000 for training and 10,000 for test), each sized 28x28 pixels. We utilize the entire test set for OOD detection.
SVHN [59] (Street View House Numbers) contains over 600,000 color images of digits captured from real-world house numbers in Google Street View. Each image is 32x32 pixels and includes significant background clutter, lighting variation, and perspective distortion. We utilize the entire test set (26,032 images) for OOD detection.
Textures [10] is a collection of texture-centric images designed for texture recognition and segmentation tasks. The dataset comprises 5,640 images from 47 texture categories, such as "bark", "brick", or "bubble", with each image exhibiting rich local patterns and high-frequency variations. We employ the entire dataset.
? Places365 [101] is large-scale scene recognition dataset comprising over 1.8 million images across 365 diverse scene categories. The standard test set includes 900 images per category. For OOD detection, we adopt the entire test set after removing 1,305 images that exhibit semantic overlap with in-distribution classes, following the filtering protocol established in prior works.
To evaluate our method under large-scale and high-resolution conditions, we adopt ImageNet-1K KK [12] as the InD dataset. This dataset comprises approximately 1.2 million training images across 1,000 diverse object categories. Following the evaluation protocol established by OpenOOD, we use 45,000 images from the official ImageNet validation set as the InD test set, while the remaining 5,000 images are held out as the InD validation set. To facilitate hyperparameter tuning and avoid information leakage from the test sets, we construct an OOD validation set using 1.763 images from OpenImage-O O [84], a curated subset of OpenImages whose categories are explicitly disjoint from ImageNet classes. For final evaluation, we benchmark our method against three widely used OOD test datasets that cover diverse semantic domains and visual characteristics:
iNaturalist [79] is large-scale, fine- ame-grained dataset containing 859,000 images of over 5,000 natural species. Captured in unconstrained environments, the images exhibit complex backgrounds, lighting variation, and high intra-class diversity, making the dataset visually and semantically distinct from object-centric benchmarks like ImageNet. Following prior works[31, 94], we use subset of 10,000 images from 110 non-overlapping classes for OOD evaluation.
Textures [10] comprises 5,640 real-world texture images spanning 47 categories. For evaluation, we employ the full dataset without further filtering.
Open Image-O [84] is curated subset of the OpenImages-v3 dataset. It comprises 17,632 images that are carefully selected to be semantically disjoint from ImageNet- classes. To ensure a rigorous evaluation setting, the subset underwent manual filtering to climinate any
33

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category overlap. Following the OpenOOD protocol, we utilize the entire dataset for OOD testing, excluding the samples reserved for OOD validation.
OOD Generalization. We adopt the training and evaluation protocol established in DomainBed [23], ensuring consistency dataset splits, training schedules, and model selection criteria. Specifically, we follow the lave-one-domain-ou evaluation strategy, where the model is trained on all but one domain and evaluated on the held-out domain to assess its generalization ability to unseen distributions. This process is repeated for each domain, treating it as the target in turn. Model selection is performed using the average accuracy across all validation sets corresponding to the training domains, ensuring that the chosen model is not tuned the target domain. We conduct experiments on five widely adopted domain generalization benchmarks included DomainBed:
PACS [45] consists of four visually distinct domains: Photo (P), Art Painting (A), Cartoon (C), and Sketch (S). It contains a total of 9,991 images spanning seven object categories common across all domains. The dataset presents significant domain shift, especially between natural (Photo) and abstract (Sketch) styles, challenging models to capture domain- invariant features.
VLCS [16] combines images from four datasets: PASCAL VOC2007 (V), LabelMe (L), Caltech- 101 (C), and SUN09 (S). contains five shared object classes and a total of 10,729 images. The main source of domain shift lies in differences in collection environments, image resolution, and labeling conventions across the constituent datasets.
OfficeHome [82] is more challenging benchmark with 15,588 images across four domains: Art (A), Clipart (C), Product (P), and Real World (R). It spans 65 object categories related to office and home environments. The dataset is characterized by high intra-class variation and large appearance gaps between stylized (Clipart, Art) and natural (Real World) domains.
Terra Inc [5] is wildlife image dataset collected from camera traps in different geo- graphical locations. It includes 24,788 images across four domains (locations): L.100, L38, L43, and L46, and contains animal species. The domain shift stems from environmental variation (e.g., lighting, terrain, vegetation) and class imbalance.
DomainNet [60] is the largest and most diverse benchmark in the domain generalization literature, comprising approximately 600,000 images across six distinct domains: Clipart, Infograph, Painting, Quickdraw, Real, and Sketch. It covers 345 object categories shared among all domains. The dataset exhibits a wide range of domain shifts, including differences in abstraction level, drawing style, color richness, and semantic representation. Among the domains, Quickdraw is particularly challenging due to its highly simplified and noisy stroke-hased illustrations.
D.3 Model Architecture
For OOD detection, we adopt post-hoc setting by freezing the classification network and training only the lightweight distribution prediction head Fa and potential prediction head dg. In contrast, for OOD generalization, all model parameters are optimized jointly.
D.3.1 OOD Detection
For the CIFAR-10 and CIFAR- 100 benchmarks, we adopt the ResNet-18 architecture [25]. In accordance with the standardized protocol in OpenOOD, we train each model for 100 epochs using the official training splits. To account for potential variability across training stages, we evaluate OOD detection performance three checkpoints saved throughout the training process. This setting ensures fair and consistent comparisons with prior post-hoc methods. Additional training configurations (e.g., learning rate, optimizer, data augmentation) follow the default settings provided in the OpenOOD.
For the large-scale ImageNet-Il 1k benchmark, we evaluate OOD detection using two represen- tative model architectures that reflect both convolutional and transformer-based paradigms: i) ResNet-50 50 [25], a 50-layer residual network pretrained on ImageNet- is employed as high- capacity convolutional backbone. During inference, all test images are resized to 224x224 pixels to match the model's input resolution. We use the official pretrained weights released by PyTorc to ensure reproducibility. ii) ViT-bl6 [15] serves as a transformer-bast based alternative, also pretrained
34

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on ImageNet- IK and evaluated with the same input resolution of 224x224 pixels. The VIT model introduces fundamentally different inductive bias compared to CNNs, allowing us examine the generality of our method across architectural paradigms. We employ the official PyTorch checkpoints for all ViT experiments.
Distribution prediction head Fg and potential prediction head Φg: For the CIFAR-I0 and CIFAR- 100 benchmarks, both the distribution prediction head Fe and the potential prediction head Φg are implemented as two-layer MLPs with an input dimension of 512, a hidden dimension of 512, and an output dimension of For the large-scale ImageNet- Ik benchmark, the hidden dimension set to 512 when using ResNet-50 as the backbone, and 1 1,536 when using ViT-b16. All heads use the SiLU activation function. Detailed configurations are provided in Table
Table 6: Details of distribution prediction head and potential prediction head.
Backhone
Input Dim.
Hiddle Dim.
Output Dim.
#ccad Param.
Activation
CIFAR-I0/100
ResNet-18 18
512
512
0.53M
SiLU
ResNet-50
2048 768
512
2.10M 2.36M
SiLU
ImageNet-Ik
VIT-h16
1536
SiLU
D.3.2 OOD Generalization
For all benchmarks, we adopt ResNet-50 pretrained on ImageNet [12] as the default backbone and initialization. Both the distribution prediction head F& and the potential prediction head Φe are implemented as two-layer MLPs with hidden dimension of 512 (same as Table 6). Although these auxiliary branches introduce a small number of additional parameters during training (approximately 8% of the total), they serve solely to guide the backbone toward learning more robust representations via explicit supervision. Importantly, these branches can be safely removed at inference time, preserving the backbone's ' structure and incurring no extra cost in prediction. Our approach maintains a clear separation between the auxiliary supervision and the primary prediction pathway, ensuring that the backbone remains unaltered throughout both training and inference. By decoupling training-time supervision from inference-time computation, achieves an efficient balance between representational robustness and deployment practicality, enabling scalable application across diverse generalization scenarios.
D.4 Hyperparameters
Shared Hyperparameters. For the constant 6o in the loss function, we uniformly assign the value 1. For each the initial velocity of the corresponding is set to vector of ones in I.Z..
OOD Detection. We train the distribution prediction head Fe and the potential prediction head de using the Adam optimizer with a learning rate of 5 x 10-4 and a batch size of 128, adhering to standard post-hoc configurations. Training proceeds for 50 epochs.
OOD Generalization. In our experiments, the wavevector in the dispersion relation loss Edisp randomly sampled from Gaussian distribution per batch, with a scaling factor of 0.5 and wavenumber N of 250. The loss weights a and are both set to 0,01. Additional hyperparameters, including learning rate, weight decay, and dropout rate, tuned following [9] and detailed in Table 3. We adopt early stopping and utilize the Adam optimizer. Consistent with DomainBed, the batch size is set to 32 for all datasets, except for DomainNet, which uses batch size of 24.
Table 7: Hyperparameters for OOD generalization experiments.
Hyperparameter
PACS
VLCCS
OfficeHome
Terralnc
DomainNet
Learning rate
3e-5
le-5
lc-5
lc-5
3e-5
Dropout
0.5
0.5
0.5
0.5
0.5
Weight decay
le-4
le-4
le-4
le-4
le-4
Training Steps
5000
5000
5000
5000
15000
35

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D.5 Experiment Infrastructure
Our experiments conducted using combination of NVIIIA GeForce RTX 3090 Ti and NVIDIA A100 GPUs. Specifically, we utilized four 3090 Ti GPUs and one A100 GPU with 40GB of memory. The OOD detection experiments are performed exclusively on the 3090 GPUs, while the OOD generalization experiments were carried out on both the 3090 Ti and A100 GPUs.
D.6 Ablation on Initial Velocity Assumption
To evaluate the robustness of our uniform initial velocity assumption, we conducted additional experiments using different initialization strategies: i) uniform vectors with all elements set to 1, ii) uniform vectors with values of 0.5 matching the feature dimension, and iii) random vectors of the same dimension. As shown in Tables 8 and 9, random initialization slightly underperforms the uniform settings, but the performance gaps are minimal, demonstrating the robustness of our approach. The uniform initialization assumption thus maintains effectiveness while simplifying computation.
Table OOD Detection results on CIFAR-100 with different initial velocity settings.
Initial Velocity
FPR95
AUROC 1
34.81
90.90
0.5 Random
35.20
90.47
38.29
89.52
Table 9: OOD Generalization results PACS with different initial velocity settingss.
Initial Velocity
Accuracy 1
1
87.7
0.5
87.7
Random
87.4
D.7 Sensitivity Analysis of MLP Architecture
We further investigate the sensitivity of our method to key architectural hyperparameters of the MLP, which serves the steady- predictor. Specifically, we examine the impact of layer depth and hidden dimension OOD detection performance using CIFAR-100 as the InD dataset. Table 10 presents results for varying layer depths with a fixed hidden dimension 512, while Table 11 shows the effect of different hidden dimensions using fixed depth of 2 layers. The results demonstrate that performance scales favorably with the number of ML.P parameters. Notably, increasing the layer depth from 1 to yields substantial improvements, with AUROC increasing from 87. 14% to 90.90% and FPR95 decreasing from 39.30 to 34.81. Similarly, expanding the hidden dimension from 128 512 progressively enhances performance. However, returns diminish beyond a certain threshold (e.g., hidden dimension 512 vs. 1024, with AUROC of 90.90% vs. 90.83% highlighting the importance of balancing model capacity with computational efficiency. These insights inform practical hyperparameter selection for deploying CBD-De in real-world applications.
Table 10: OOD detection on CIFAR-100 with MLP hidden dimension of 512.
Layer Depth
FPR95 +
AUROC +
39.30
87.14
2
34.81
90.90
36

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Table 11: OOD detection on CIFAR-100 with MLP depth of 2 layers.
Hidden Dimension
FPR95 L AUROC
128
40.35
86.93
256
38.52
88.20
512
34.81
90.90
1024
35.25
90.83
D.8 Validation of the MLP Steady-State Approximation
Our formulation of staad prediction is originally defined through PDE. To make the solution computationally tractable, we replace the numerical PDE integration with an MLP-based steady-state predictor that directly approximates the stationary solution single forward pass. To assess the fidelity of this approximation, we conduct experiments using numerical PDE solver implemented with DeepXDE [53], which follows the same architecture and parameter settings (2 layers, 512 hidden dimensions). The PDE solver is executed with 10 and 30 integration steps to estimate the steady-state distribution. Table 12 presents the OOD detection results on CIFAR- 10 and CIFAR- 100 datasets. For simpler datasets (CIFAR-10) performance differences are negligible (0.02-0.6) AUROC), while for more complex datasets (CIFAR-100, numerical integration shows moderate improvements (0.30-0.42 AUROC). However, these gains require 7-23x longer inference time and substantial memory verhead for intermediate computations. Given the minimal accuracy differences on simpler distributions and the significant computational advantages, direct prediction approach offers practical solution for efficient OOD detection in real-world applications.
Table 12: Comparison of the MLP steady-state approximation against a numerical PDE solver implemented with DeepXDE [53]. AUROC and relative inference time are reported.
Dataset
Sample Size
Classes
MLP AUROC
PDE Solver AUROC
Forward Pass Time
Time Steps = 10
CIFAR-10 10
50,000
10
96.39
96.41 (+0.02)
7x slower
CIFAR-100
50,000
100
90.90
91.20 (+0.30)
7x slower
Time Steps 30
CIFAR-10
50,000
10
96.39
96.45 (+0.06)
23x slower 23x slower
CIFAR-100
50,000
100
90.90
91.32 (+0.42)
E Full OOD Detetion Results
37

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Table 13: OOD detection per formaaacc on the CIFAR- 10 benchmark. Results are formatted as first, second, and third best. Following the OpenOOD protocol, we report the average
AUROCT
89.6220.18 19
90.7310.45
90.97+0.52
87.961061
89.7211.36
73.90=0.1
92.20
71.73
91.21
54.61#1551
57.55+322
90.42
91.1010.00
82.68=0.11
93.48+0.24
92.96
84.23
75.87
78.49+2.55
85.32+1.45
91.35
94.60=0.4
96.39 96.39=0.3
+0.81
Average
93.
78.
29.69
31.72
$3.48#2331
57.62
32.221340
61.47=048
25.35=073
1673
41.69
83.52211.68
91 11.90=223
44 24.9
41 41.68151 1527
78.31
25.05
24.27
51.7622442
$7.44=7.00
79.03+4.22
66 66.4825.90
34.73
18.31
17.65 1..65
FPR951
57.
AUROCT
58.6311028
88 08.9244047
89. 89.11+0552
85.07
84.901254
52.4740.15
91.51+0011
60. 60.44.1341
89.251008
53.3415.79
60.50+338
90.3510078
89. 19.141076
78.374033
89 .9.49999
4-22
74.6744.95
85 1..941 9:904
79. 49.8993.6
82. 4:122
89 19.4610065
41.85
93.181073
Places365
38.52+22.7
42,47
45.27+4450
70.36.:6.6
47.6714.54
94.16:00.3
31.20#20.8
90.4911.93
54.851652
95.3214445
92.46++226
44.2013..5
54.8416.51
95.58+2.12
41.43#2.1
30.38
77.19
43.70
77.89+7.28
76.36.15.22
47.03
26.73++0.0
22,75
FPR95
56.14118.26
AUROC1
89. 10.60
89.89
90.01 +0.74
86. 36.94112.0
92.6911.06
77.40=0.28
92.23
62.34
89.47
52.07
89.3811.40
89.41 10.71
82.35
95.15
93.1611024
81 1..8612.35
73.4616449
77.45+2.30
8 11.57
90.1410.76
95.64+0044
97.03
Textures
CIFAR 10 Benchmark
89
31. 1..50
34. 96.96644664
38. 16::5.89
67. 70.0
27. 7..4444.20
16.07=00.17
25.25 25.255
94. 14.44122.71
51. 12.82.6111
83.95 13.9551144.8 14.9
98. 09:404499
51. 12
51.73
81. 81.95+-9.5
21.14:11.3
24.06:10.55
62. 42.4..4779
59. 71
84. 19.47.174
84. 60.5:353300
40. 00.74166.11
17.03+0.59
18. 18.44411.2
FPR951
performance over three ResNet-18 18 checkpoints, each trained exclusively on the InD dataset.
AUROCT
10.47
91,46110.0
91.66++052
84.58 10.77
91.1811047
66.56+4058
91.84++0.26
91.52
91.7910.98
52.81
53.91+6.36
89. 19.1213.19
91.69 10.94
84,99+1
94.50+0.0
92.67 14.30
90.02+1.77
68. 151744
73 73.46+6641
86.38
91.97 10.66
96.05 400.47
97.85
SVHN
FPR95
25. 1.47
25 15.82+1664
26.97+265
68.61
25.961252
74.34=104
23 23,46=148
33.91
35.12
75.27=2693
91.65+242
50.23 15.98
35.09 1609
59 7700
19.2 2041
22. 12.
36.6 36.61 14.74
64.49 44.49
83.64+6.18
62.74
28.14 28.1442559
14.33
14.22=1500
AUROCT
04.4
92.63+157
93.11+1777
1.99
90. 10.101241
99.1
93. 93,22+080
72.64
94.324258
56.14422400
08
63.72+7.77
94.151248
85.00+204
94.7620038
94.26+1038
90.374559
75.87+522
83. 16+4666
4.443
93.831214
94.86+1.30 36
97. 17.48+122
92.81
85.0
MINIST
FPR951
23.33 23.33
23.641581
23.53
23.83 1233
27.301355
1.30=051
21.49 4-232
70.30 8.96
1299
24.90 99.94:: 1298
79.54 19.71
85 44.85
33. 1
25 11187
76.22=1200
1.46
20.05 11.36
10.44
61. 12.78
10.56
42. 12.22+200 20.59
23.0
-2.00
0..20=0399
Max
TempScale
MDSEnS
OpenGAN
GradNorm
RankFeat
CBD-De
Method
MSP
ODIN
MDS
RMDS
Gram
EBO
ReAct
MLS
KLM
VIM
KNN
DICE
ASH
SHE
GEN
NAC-UE
38

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Table 14: OOD detection performance on the CIFAR- 100 henchmark. Results are formatted as first, second. and third best. Following the OpenOOD protocnl, we report the average
AUROC1
79. 10.41
77.76:1.44
78. +..54
79.28 79.2810.21
69. 19.3911.39
66.0040.60
82.92.1.42 10.42
73.3641.00
79.77
67.8817.16
69. 14.10.0
80.39+0..9
79.6710.57
76.24+0.52
81. $1.7010.22
82 $2.40+0.17
80.01
67. 17.1222.2
80.581066
76.92+1.16
90.9010.66
79.68
86, 16.98+00
Average
FPR95,
54.50=06.
58. 58.70 75.70....00 1.06
57. 97.94+11 1:14
58.86=0.79
72.26
66.74=1004
52.811063
1063
64.44
56.59
70.49 17.38
83.68 18.92
54.20=1.56
56.73
71. 71.65
50.74
53.65 +0028
56.2511061 1000
69. 45
59. 20
64.12+2m
56.71
40. 10.14++.86
34,81
52.81
56.25
AUROC1
79. 19.299:0000
79.22 .9.222+0.99
79 40
79.4 79.45+-0.26
63.15
42.27++0.3
83.40 10.46
46.38+1.21
79.5240.25
69. 69.13
69.6910.17
80.03++0.1
79.7511024
75.70+0.24
75.85 75.8510.37
09.43++0.47
78.33
63.82+1.85
78.7610.16
76.30
80.28
81 $1.50
1616
73.05
Places365
FPR95.
54.85 1.44
56.62=0.87
66.46=000
-094
59.7 =0.92
79.61 =0.34
96.61 =0.17
10.43
=0.60
57.75-0.5
73.75=8.31 -
85.32-0.44
55 55.30=0.41
57.68 -091
81.40=1.50
61.57-0.77 50.70.7
60.70=1.03
59.39
77.42=1.66
62.95
65.24
36.25=100
13.57+1.16
55.23
53.57
56.
56
AUROCT
00.5
77.32
78. 1:177
76. 26.26.
69. 11.14
83.65 13.651051 1051
11.7921.32
78.35
65. 23.44
64.58 (4.5810133
80. 1044
18.39.084
85.91 1078
83 66:40.8
77. 7.34
69. 40:
80.72 00.72.1070
73.64
78. 20
99.32+0055
90.02
79.33
75.77
Textures
CIFAR- 100 Benchmark
FPR95.
56. 16.1911191
61.88 11.88.1.22 1.28
61.56.11.45
62.37
70.4912.48
84.94+0.8
53.99 $3.99.1.1
89.5 19.51
62.354206
74.77
92.51
55.04+0.82
62.39.2.13
81.8045.00
46.8612.29
$3.56+22.32
64.231166
06.87+3.80
61.271274
73.29
61 11.2311140
40.65+11.4
37.3412.30
performance Over three ResNet-18 18 checkpoints, each truined exclusively on the InD dataset.
AUROCT
11.53
78.4 =8.42
79.79=1.05
74.54
26.48
53.76=166
84.89 11.10
93.55
82.03
68. 18.40=2215
76.95
83 =0991
1..49
09.34
83.14_3.1
84.15=1009
84.22
72.14=1.30
85.60=1.40
80.97=3.98
81 11.51
92 1..40+1.26
95.80=2.5
79.34
SVHN
FPR95
53. 1.720
22.53
57.71
67.41 13.88
1600 1609
$2.57=258
51.65 13.68
20.06
53.62 23114
7-.35-2206
7.94
50.41=200 =202
13004
50.30 7.04
46.22
51.75+3112
49.58 1332
58 +2.4
46. 1266
59.15
55.45 2276
24.39+4.66 44.66
18.72 16 12
50.41
53.90
69.
AUROCT
76.01 1:339
76. 1.88
83. $3.7921131
67.4 20.81
98.21+0078
79. 14.492249
+4.15
79. 18
68. 118778
65. 1.1211112
78. 1.5
1147
74. 15+259
81. 19.991002
2.366+152
79. 19.8611.89
63.03 33.03333.6
17.234046
76. 767
78. 29.05
1.66
96. 19
MINIST
FPR95
53 43.82+4.74
14.68
56.05+4661
45.9 45.9413.29
71.72
-.83
52.05 52.0516.28 16.28
53.53+745
52.624383
63.00 23.09=23.25
36.9711.44
56.0445.66
52.95 13382
73 73.09+667
18.32
+4.67
51.7913.67
75.01
13.88
58. 58.78+21 +2.78
+2.70
53.9215.71
2669
47.93
OpenMax
TempScale
MDSEns
OpenGAN
GradNorm
RankFeat
NAC-UE
CBD-De
Method
MSP
ODIN
MDS
RMDS
Gram
EBO
ReAct
MLS
KLM
VIM
KNN
DICE
ASH
SHE
GEN
39

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Table 15: OOD detection performance on the ImageNet- 1k benchmark, Results are formatted as first. second, and third best. Following the OpenOOD protocol, we report the Vit-b16 backbones, each trained exclusively on the InD dataset.
Average
86.81
83.75
85.17
89.00
89.61
79.94
87.73
88.02
84.94
68.52
89.73
86.07
85.61
94.29
94.11
87.13
70.90
72.72
93.13
88.91
96.04
96.31
Vit-b16
Textures
85.52
85.06
85.39
89.41
89.38
81.17
44.99
86.66
83.74
86.49
90.61
91.12
82.21
48.53
92.65
90.23
94.17
94.30
/
/
ResNet-50
58.10
82.43
84.95
89.00
89.80
79.94
86.08
88.02
88.70
92.05
92.79
88.39
84.72
97.97
97.09
92.04
70.90
96.90
93.60
87.59
97.90
98.31
/
Average
87.49
84.86
86.13
88.23
80.83
60.80
89.08
74.43
82.77
61.32
88.08
85.39
87.17
91.34
88.45
85.24
50.83
74.39
88.78
89.77
91.52
92.27
Openlmage-O
Vit-b16
87.36
84.86
85.04
92.38
92.32
76.48
37.82
84.29
81.60
87.03
92.18
89.86
82.22
55.51
91.04
90.27
91.58
91.89
/
1
ResNet-50
87.62
84.86
87.22
88.23
69.27
60.80
85.84
74.43
89.06
84.82
91.87
89.17
87.30
90.50
87.04
88.26
50.83
93.26
86.52
89.26
91.45
92.65
/
Average
93. 49
88.30
89.52
91.17
79.84
61.82
91.67
76.67
84.97
68.16
91.23
88.23
90.19
92.64
88.94
87.52
40.06
73.85
93.11
92.99
95.12
97.16
iNaturalist
Vit-b16
94.93
88. 19
88.54
96.01
96.10
79.30
42.42
86.11
85.29
89.59
95.72
91.46
82.50
50.62
93.57
93.54
93.72
97.26
/
average performance over ResNer-50 and and
ResNet-50
92.05
88.41
90.50
91.17
63.67
61.82
87.24
76.67
90.63
93.89
96.34
91.17
90.78
89.56
86.41
92.54
40.06
97.07
92.65
92.44
96.52
97.06
/
OpenMax
TempScale
MDSEns
OpenGAN
GradNorm
RankFeat
NAC-UE
CBD-De
Method
MSP
ODIN
MDS
RMDS
Gram
EBO
ReAct
MLS
KLM
VIM
KNN
DICE
ASH
SHE
GEN
40

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Full OOD Generalization Results
Table Out-of-domain accuracies (%) on PACS. Results are formatted as first, second, and third best.
Method
A
C
P
S
Avg
IRM
84.8+1.3
76.4+1.1
96.7=0.6
76.1
83.5
ERM
85.7 10.6
77.1 =0.8
97.4 10.4
76.6 10.7
84.2
GroupDRO
83.510.9
79.1
96.710.7
78.312.0
84.4
MTL
87.5+0.8
77.1
96.4=0.8
77.3+1.8
84.6
Mixup
86.1
78.9
97.6-0.11
75.8+1.8
84.6
MMD
86.111.4
79.410.9
96.610.2
16.5
84.7
VREx
86.0+1.6
79.1
96.9
77.7+1.7
84.9
MLDG
85.5+14
80.1
97.410.3
76.6+1.1
84.9
ARM
86.810.6
76.810.5
97.4
79.311.2
85.1
Mixstyle
86.8+0.5
79.0#1.4
96.6-0.1
78.5+2.3
85.2
CORAL
88.3+0.2
80.0+0.5
97.5
78.8+1.3
86.2
SagNet
87.410.2
80.7
97.1
80.0
86.3
RSC
85.4+0.9
79.7
97.6+t0.9
78.2+1.0
85.2
SAM
85.6+2.1
80.9+1.2
97.0=0.4
79.6+1.6
85.8
GSAM
86.910.1
80.410.2
97.5
78.710.8
85.9
SAGM
87.4
80.2+0.3
98.0=0.2
80.8+0.6
86.6
GGA
88.8+0022
80.1
97.3+0.2
81.2+00.5
87.3
CBD-Gen
89.0+0.6
81.6+0.4
98.1+0.3
82.1+0.8
87.7
Table 17: Out-of-domain accuracies (%) on VLCS, Results are formatted as first, second, and third
best.
Method
C
L
S
V
Avg
GroupDRO
97.3+0.3
63.440.9
69.5
76.7+0.7
76.7
MLDG
97.410.2
65.210.7
71.011.
75.311.0
77.2
MTL
97.8+0.4
64.3+0.3
71.3+0.7
75.3+1.7
77.2
ERM
98.0 40.3
64.7 =1.2
71.4 41.2
75.2 +1.6
77.3
Mixup
98.310.6
64.811.0
72.1
74.310.8
77.4
MMD
97.7+0.1
64.0+1.1
72.8+0.2
75.3+3.3
77.5
ARM
98.7+0.2
63.610.7
71.3+1.2
76.740.6
77.6
SagNet
97.910.4
64.510.5
71,411.3
77.5
77.8
Mixstyle
98.640.3
64.5
72,6
75.7
77.9
VREx
98.4+0.3
64.4+14
74.1
1..
78.3
IRM
98.610.1
64.910.9
73.4
77.3
78.6
CORAL
98.3+0.3
66.1
73.4
77.5
78.8
RSC
97.9+0.1
62.5+0.7
12.
75.6+0.8
77.1
GSAM
98.710.3
64.910.2
74.3
78.5
79.1
SAM
99.1
65.0+1.0
73.7+1.0
79.8
79.4
SAGM
99.0+0.2
65.2+0.4
75.1
80.7+0.8
80.0
GGA
99.110.2
67.510.6
75.1
78.010.1
79.9
CBD-Gen
99.1+0.2
67.3
75.1
80.5+1.0
80.5
41

---

Table 18: Out-of-domain accuracies (%) OfficeHome. Results formatted as first, second, and
third best.
Algorithm
A
C
P
R
Avg
Mixstyle
51.1
53.2
68.2
69.2
60,4
IRM
58.912.3
52.2.1.6
12.
7..012.5
64.3
ARM
58.910.8
51.0+0..5
74.1
75.2
64.8
GroupDRO
60.4+0.7
52.7
75.0=0.7
76.0+0.7
66.0
MMD
60.410.2
53.310.3
74.3.001
77.410..
66.4
MTL
61.5+0.7
52.410.6
74.9
76.8+0.4
66.4
VREx
60.7
53.0+0.9
75.
76.6+0.5
66.4
ERM
63.1
51.9 10.4
77.2 10.5
78.1 10.2
67.6
MLDG
61.5+0.9
53.2+0.6
75.0=1.2
07.
66.8
Mixup
62.4+0.8
54.8+0.6
76.9
78.3+002
68.1
SagNet
63.4+0.2
54.8+0.4
75.
78.
68.1
CORAL
65.3+0.3
54.4+0.6
76.5
1..4+1.0
68.7
RSC
60.7+1.4
51.4+0.3
74.8+1.1
1..
65.5
GSAM
64.9+0.1
55.2+0.2
77.8+0.0
79.2+0.2
69.3
SAM
64.5+0.3
56.5+0.2
77.4
79.8+0.4
69.6
SAGM
65.4+0.4
57.0+0.3
78.0=0.3
80.0+0.2
70.1
GGA
64.3+0.1
54.4+0.2
+0.3
78.9+0.2
68.5
CBD-Gen
66.5
58.7
78.8
81.6
71.4
Table 19: Out-of-domain accuracies (%) on TerraInc. Results are formatted first, second, and third
best.
Method
L.00
L38
I.43
I.66
Avg
MMD
41.913.0
34.811.0
17.0
35.2
42.2
GroupDRO
41.210.7
38.612.1
56.7
36.412.1
43.2
Mixstyle
54.3+11.1
34.1+1.1
55.9+1.1
31.742.1
44.0
ARM
49.310.7
38.310.7
55.8
38.711.3
45.5
MTL
49.311.2
39.616.3
55.6
37.8+008
45.6
ERM
49.8 44.4
42.1 114
56.9 11.8
35.7 43.9
46.1
VREx
48.214.3
41.7
56.8
38.713.1
46.4
IRM
54.611.3
39.8+1.9
56.211.8
39.6+0.8
47.6
CORAL
51.6+2.4
42.2+1.0
57.0=1.0
39.8+2.9
47.6
MLDG
54.213.0
44.311.1
55.610.3
36.912.2
47.8
Mixup
59.6+2.0
42.2+1.4
55.9
33.9+1.4
47.9
SagNet
53.0+2.0
43.0+1.4
57.9+0.8
40.4+1.4
48.6
SAM
46.311.0
38.412.
54.0.1.0
34.5
43.3
RSC
50.2+2.2
39.2+1.4
56.3+1.4
40.8+0.6
46.6
GSAM
50.8+0.1
39.3+0.2
59.6+0.0
38.2+0.8
47.0
SAGM
54.8+1.3
41.4+0.8
57.7+0.6
41.3+0.4
48.8
GGA
55.9
45.5
59.7+0.1
41.5+0.1
50.6
CBD-Gen
56.4+2.1
45.0+1.2
59.6+0.6
41.0+1.5
50.5
42

---

Table 20: Out-of-domain accuracies (%) on DomainNet. Results formatted as first, second, and third best.
Algorithm
clip
info
paint
quick
real
sketch
Avg
MMD
32.1 +13.3
11.0 #4.6
26.8 +113
8.7 #2.1
32.7 #13.8
28.9 #11.
23.4
GroupDRO
47.2 40.5
17.5 +0.4
33.8 =0.5
9.3 +0.3
51.6 =0.4
40.1 10.6
33.3
VREx
47.3 43.5
16.0 11.
35.8 #4.6
10,9 #0.3
49.6 #4.9
42.0 #3.0
33.6
IRM
48.5
15.0
38.3 +4.3
10.9 20.5
48.2
42.3 13.1
33.9
Mixstyle
51.9 400.4
13.3 #0.2
37.0 #0.5
12.3 20.1
46.1 #0.3
43.4 20.4
34.0
ARM
49.7
16.3
40.9 11.1
4..
53.4
43.5 10.4
35.5
Mixup
55.74003
18.5+0.5
44.3
12.540.4
55.820.3
48.2
39.2
SagNet
57.7
19.0
45.3 10.3
12.7 10.5
58.1
48.8 10.2
40.3
MTL
57.9 10.5
18.5 10.4
46.0 10.1
12.5 10.1
59.5 10.3
49.2 10.1
40.6
MLDG
59.1 10.2
19.1
45.8 10.7
13.4 10.3
59.6
50.2 10.4
41.2
CORAL
59.2 10.1
19.7 10.2
46.6 10.3
13.4 10.4
59.8 10.2
50.1 10.6
41.5
ERM
63.0 10.2
21.2
50.1 10.4
13.9 10.5
63.7 10.2
52.0 10.5
43.8
RSC
55.01112
18.3
44,410.6
12.210.2
55.7
=7.8
38.9
SAM
64.5+0.3
20.7+0.2
50.2+0.1
15.1+0.3
62.640.2
52.7=0.3
44.3
GSAM
64.2+00.
20.8+0.2
50.9+0.0
14.4+0.8
63.5+0.2
53.9
44.6
SAGM
64.9+0.2
21.1+0.3
51.540.2
14.8+0.2
64.1+0.2
53.6-0.2
45.0
GGA
64.0+0.2
22.2+00.3
51.740.1
14.3+0.2
64.11004
54.3
45.2
CBD-Gen
64.9
22.8
52.540.3
15.1
64.7-10.4
55.4
45.9
43

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Reviews:
[{"review_id": "q6Ehz076sU", "reviewer": "Reviewer_MQNz", "summary": "The paper introduces a novel physics perspective on out-of-distribution detection by treating high-level data features as particles in a field. These particles evolve under the influence of a learned potential field, with in-distribution samples converging toward a steady-state region or “basin of attraction.” The boundary of this basin is then used as a decision surface for detecting OOD inputs. The authors support their proposal with theoretical results involving partial differential equations and evaluate their approach on standard OOD benchmarks.", "questions": "Please help me address the weaknesses above. In particular, can you explain the full training and testing procedure for your models?\nAdditionally, what hyperparameter tuning was performed for your method? (e.g. how did you choose the MLP architectures etc.)\n\nNote that I am willing to raise my score upon reaching more clarity on the above.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 4, "strengths_and_weaknesses": "*Strengths*\n- The idea of interpreting feature dynamics via a physical system governed by a PDE and identifying the basin of attraction as an OOD boundary is original and interesting\n- The authors attempt to rigorously ground their model in physics-inspired PDE theory, including steady-state solutions and stability analysis \n- The approach appears to be usable in a post-hoc setting, potentially making it broadly applicable to existing trained models (though this is not clearly communicated in the main text).\n- The authors speculate that model robustness may correlate with the size of the basin of attraction, which, if substantiated, could offer a new direction for designing new methods.\n- The approach achieves supposedly SotA performance for OOD detection and generalisation (however, I was not able to reach sufficient clarity on how this method produces scores for an individual input point, and how exactly the networks are trained to confidently interpret these results, see weakness below)\n\n*Weaknesses*\n- Key terms like \"collective behavior,\" \"self-consistent electric field,\" \"macroscopic properties,\" and \"perturbation\" are not clearly defined early in the paper, making it very hard to appreciate clearly without several passes\n- Many technical statements are vague or confusing (e.g., “Plane wave modes” as an x-axis label in a figure; unclear phrasing such as “measure its boundary” without formally clarifying what \"its\" refers to).\n- The flow of theoretical principles (e.g., Principles 1 and 2) lacks sufficient intuitive grounding and logical coherence (i.e. \"thus\" is used despite it being unclear how the two principles are necessary or sufficient for the subsequent statement).\n- Several figures (e.g., Figure 1 and Figure 2) are confusing or potentially mislabeled, weakening the explanatory power of the paper.\n- There is an over-reliance on jargon without providing accessible interpretations early in the paper, making the work difficult to follow even for an expert audience familiar with OOD literature.\n- It’s not transparent from the main body how the method is actually used to compute OOD scores. Only by reading the appendix does it become evident that the method is post-hoc (\"post-hoc\" is not even mentioned anywhere in the paper, nor the fact that the F and phi functions are learned with gradient descent using held out InD data; my experience of reading this paper was that I like the idea but I felt continuously unclear what the method itself actually is until reading the appendix--I should not need to read the appendix to appreciate the actual algorithm)\n- The use of \"batch\" in the pseudocode is ambiguous -- it's not clear whether the method assumes access to a test-time batch or operates on individual points, it's not clear whether the batch is used just to fit the F and phi, but then at test time use only a singleton input point, or if the intended use is to use batches at test time.\n- “we posit that network robustness improves as the basin’s range expands.” → “How can we enlarge basin of attraction’s boundary to enhance model generalization?” This is an interesting idea but I don’t like that they went from an idea to a question that presumes the idea is good and operationalises it. It is not clear that what they posit is indeed true, so it feels like a big leap to then start already seeking to maximize it.\n- Notation in Theorem 2 (e.g., $L_1$, $L_\\infty$, $n_i(z)$, $\\delta(v−v_0)$) is introduced without definition.\n- The physical analogy (particles, mass, steady state) is inconsistently applied and appears more illustrative than foundational e.g. (\"mass\" appears once in the paper, but no definition or intuition).\n- Modulo my lack of clarity on the exact evaluation setting (e.g. what is the \"batch\" in the pseudo-code, how many particles lie in R^z for an individual input point (is this $N$?)) the results seem compelling, but confidence intervals are provided in the tables without any explanation for what they represent and how they are constructed.\n\n*Small points*\n- Figure 1 shows only a single particle, despite the paper’s emphasis on collective particle dynamics. Consider revising to reflect the collective behavior.\n- The x-axis label of Figure 2 “Plane wave modes” is confusing and lacks explanation. Also, fix the typo “solusion.”\n- In the sentence “as illustrated in Figure 6,” it seems you meant Figure 2 instead.", "was_revised": true, "final_justification": "I found this paper to be interesting, although limited in empirical rigor and clarity. I am somewhat concerned that the paper is excessively \"motivated\", in the sense that the authors seem like they want the physics interpretation to be constructive even if the empirical evidence of its value is less clear. However, the authors engaged with me and addressed most of my clarity concerns during the rebuttal, and with the updates I think the paper is sufficiently interesting to justify my recommendation.", "rebuttal": "Thank you for the positive feedback and insightful questions. We really appreciate your kind words about our work, and your suggestions will help us improve both the quality and clarity of our research. Below, we will address your concerns one by one.\n\n---\n\n> Q1: Key terms like... are not clearly defined early in the paper, making it very hard to appreciate clearly without several passes.\n\nWe acknowledge that key terms lack clear definitions early in the paper. In the revision, we will include the following definitions in the Introduction:\n\n- **Collective behavior:** Emergent system-wide phenomena resulting from multi-particle interactions. In our work, where high-level features are treated as charged particles, this represents the temporal evolution of feature distributions driven by inter-feature interactions.\n- **Self-consistent electric field:** Field where particles generate the electric field that governs their motion, creating iterative dynamics. Particles produce a field, which drives their movement, redistributing the particles and generating a new field until equilibrium is reached, leading to collective behavior.\n- **Macroscopic properties:** System-level characteristics, which in our work refer to the particle distribution function $F(\\mathbf{z},\\mathbf{v},t)$.\n- **Perturbation:** Minor deviations from the steady-state particle distribution function and electric potential (line 161).\n\n---\n\n> Q2: Many technical statements are vague or confusing (e.g., \"Plane wave modes\"... what \"its\" refers to).\n\nWe acknowledge that several technical statements require clarification:\n\n- **Regarding the x-axis label in the Figure 2:** It should be labeled 'time'. Plane wave modes decompose perturbations into constituent waves with distinct directions and wavelengths, similar to how Fourier transforms decompose images into frequency components, revealing each wave's contribution to the overall perturbation.\n- **Regarding unclear pronoun references:** In the expression 'measure its boundary', the pronoun 'its' specifically refers to the 'basin of attraction'. \n\nWe will conduct a thorough review of the manuscript and revise any instances of ambiguous words in the revised submission.\n\n---\n\n> Q3: The flow of theoretical principles... lacks sufficient intuitive grounding and logical coherence.\n\nWe sincerely apologize for the unclear logical progression of our theoretical principles and thank you for this valuable feedback. We respectfully encourage you to refer to our response to Reviewer $\\text{\\textcolor{red}{jZnV}}$'s question $\\text{\\textcolor{blue}{Q1}}$ for the motivation behind our modeling approach. Building upon that foundation, the relationships among our theoretical principles follow this logical sequence:\n\n- Theorem 2.1 establishes that steady-state solutions of Vlasov-Poisson systems provide a theoretical foundation for OOD detection.\n- Since directly solving the steady-state electric field $E^* (\\mathbf{z})$ (Eq 4) involves computationally challenging integrals, Definition 2.2 introduces a tractable condition via the Poisson-Boltzmann equation for the steady-state electric potential $\\phi^* (\\mathbf{z})$, simplifying the computation of $E^*(\\mathbf{z})$.\n- Corollary 2.3 connects the basin of attraction $\\mathcal{B}$, its boundary $\\partial \\mathcal{B}$, and the steady-state potential $\\phi^*(\\mathbf{z})$, revealing sharp gradients near $\\partial \\mathcal{B}$ that further justify the system's applicability to OOD detection.\n- Theorem 2.4 proves that our proposed dispersion loss expands the basin of attraction $\\mathcal{B}$.\n- Theorem 2.5 and Corollary 2.6 demonstrate that this expansion enhances neural network robustness.\n\nWe will clarify these logical connections with appropriate transitional explanations in the revision.\n\n---\n\n> Q4: Figures errors (with all small points).\n\nThank you for the detailed observations! We will revise all relevant figures in the revision\nto correct labeling errors, improve axis descriptions, fix spelling errors, and ensure the figures accurately reflect the collective dynamics discussed in the text.\n\n---\n\n> Q5: There is an over-reliance on jargon.\n\nWe apologize for not adequately considering the accessibility of our mathematical and physics concepts for readers with diverse backgrounds. To improve clarity, we will add accessible explanations in the introduction footnotes, and provide intuitive descriptions of technical terms in the appendix. We hope these revisions address your concerns.\n\n---\n\n> Q6: It's not transparent from the main body how the method is actually used... the method is post-hoc... I like the idea but I felt unclear what the method itself actually is until reading the appendix...\n\nWe thank you for appreciating the core idea. We acknowledge that the main text inadequately explained the post-hoc nature of our method and the OOD implementation process. In the revised manuscript, we will:\n\n- Explicitly present the OOD detection loss $\\mathcal{L}_ {\\text{De}} = \\mathcal{L}_ {\\text{Vlasov}} + \\mathcal{L}_ {\\text{Poisson}}$ and OOD generalization loss $\\mathcal{L}_ {\\text{Gen}} = \\mathcal{L}_ {\\text{Classify}} + \\alpha(\\mathcal{L}_ {\\text{Vlasov}} + \\mathcal{L}_ {\\text{Poisson}}) + \\beta \\mathcal{L}_ {\\text{disp}}$ in the methodology section.\n- Clarify in the Introduction that *CBD*-De is a post-hoc detection method that does not require modifying pre-trained network parameters.\n- Label the input as InD data in the left subplot of Figure 3 to clarify that $F$ and $\\phi$ are learned on the InD data.\n\n---\n\n> Q7: The use of \"batch\" in the pseudocode is ambiguous...\n\nThe batches in our pseudocode are used exclusively during training to fit $F^* (\\mathbf{z}, \\mathbf{v})$ and $\\phi^* (\\mathbf{z})$, while at test time we process **singleton** inputs (or **arbitrary** batch sizes). We will clarify this distinction in the revised manuscript.\n\n---\n\n> Q8: ... from 'we posit...' to 'how can we enlarge...' may unintentionally suggest that the hypothesis is taken as established fact.\n\nWe acknowledge this logical leap and thank you for this important observation. The transition from positing our hypothesis ('robustness improves as the basin's range expands') to immediately asking how to enlarge the basin was premature, as it assumed the hypothesis was valid. Our intent was to explore a promising direction, not assert a fact. While **Theorem 2.5 and Corollary 2.6 provide theoretical support for this hypothesis**, we will revise the manuscript to explicitly frame basin enlargement as a hypothesis-driven investigation.\n\n---\n\n> Q9: ... $L_1$, $L_\\infty$, $n_i(z)$, $\\delta(v−v_0)$ is introduced without definition.\n\nThank you for highlighting the missing definitions in Theorem 2.1. This was an oversight on our part. In the revision, we've added the following explanations:\n\n- $L^1(\\mathbb{R})$ refers to the space of integrable functions on $\\mathbb{R}$, while $L^\\infty(\\mathbb{R})$ denotes the space of essentially bounded functions.\n- $n_i(z)$ represents the spatial density for the $i$-th system.\n- $\\delta(v - v_0)$ is the Dirac delta distribution centered at $v = v_0$, indicating a velocity distribution focused at that point.\n\n---\n\n> Q10: ... (particles, mass, steady state) is inconsistently applied and appears more illustrative than foundational...\n\nThat's a great question. It gives us a chance to elaborate on our thinking about neural networks, especially the concept of \"mass\".\n\n- In physics, particles have position. It's reasonable to use \"particles\" for advanced features because they behave like **discrete entities** in a high-dimensional feature space, akin to charged particles in plasma driving dynamics similar to network training.\n- \"Mass\" relates to inertia—the property that allows an object to maintain its state of motion (rest or constant velocity), with greater mass requiring more force to induce change. In neural networks, **we can interpret \"mass\" as a feature's \"inertia\" or \"stability\"**, so features with higher mass resist shifts during learning or optimization, much like mass influences trajectories in physics. This could link to the feature's impact on network output.\n- As for \"steady state,\" it corresponds to the condition where the partial derivative of the distribution function $ F(\\mathbf{z}, \\mathbf{v}, t) $ with respect to time equals zero, indicating equilibrium.\n\nWe’ll include dedicated sentences in the revision to expand on these ideas.\n\n---\n\n> Q11: Modulo my lack of clarity on the exact evaluation setting...\n\nThanks for the feedback. The batch definition is addressed in our response to $\\text{\\textcolor{blue}{Q7}}$. Confidence intervals in the tables reflect mean ± standard error over three independent test runs. We'll clarify this in the table captions.\n\n---\n\n> Q12: Full training and testing procedure and hyperparameter tuning.\n\n- **For OOD detection:** We freeze a pre-trained model and add two MLP branches after the feature extractor (Figure 3), training only these branches on InD data with loss $\\mathcal{L}_{\\text{De}}$ and batch size 128. At test time, scores are computed from MLP outputs (Eq.9) with flexible batch sizing.\n- **For OOD generalization:** We train all model parameters and  MLP branches with loss $\\mathcal{L}_{\\text{Gen}}$ using batch size 32 (24 for DomainNet). Figure 6 provides batch size analysis. Testing uses classifier outputs only, with arbitrary batch sizes supported.\n- **Hyperparameter tuning:** We employed Optuna with systematic optimization based on GPU memory constraints and domain knowledge from related OOD literature. Using TPE sampling across 30 trials, each configuration was evaluated with 3 random seeds for stability. For example, the search space on CBD-De included:\n\n```python\n{\n    'batch_size': [32, 64, 128],\n    'learning_rate': [1e-3, 5e-4, 1e-4, 5e-5], \n    'hidden_dim': [128, 256, 512, 1024],\n    'MLP_layers': [1, 2],\n    ...\n}\n```\n\nAll reported results use the optimal configurations identified through this process."}, {"review_id": "mYBcNKpTZK", "reviewer": "Reviewer_KKbs", "summary": "This paper introduces a novel physics-inspired framework named Collective Behavior Dynamics (CBD) to tackle out-of-distribution (OOD) problems. The core idea is to treat high-level features from a neural network as charged particles interacting within a self-consistent electric field, and collective behavior of these particles is modeled using the Vlasov-Poisson system. In-distribution data is assumed to correspond to initial states that converge to a steady state within this system, and the set of all such initial states forms a \"basin of attraction\". OOD detection is then framed as determining whether a new input falls within this basin. For OOD generalization, the goal is to expand the basin of attraction, making the model robust to a wider range of perturbations. The authors achieve this by analyzing system stability through a dispersion relation and introducing a regularization loss to suppress unstable modes.", "questions": "Regarding In-Distribution Structure: The mathematical implementation of the paper (Equation 4-8) seems to build only one steady-state solution and one corresponding basin of attraction. This presupposes that the dynamic behavior of in-distribution (In-D) data is unimodal. However, real-world datasets are often highly multimodal. How to bridge this gap?\n\nRegarding the Model Choice: How crucial is the specific choice of the Vlasov-Poisson system? Could simpler dynamical systems or energy-based models capture the essential dynamics of InD vs. OOD data?\n\nRegarding steady state: The entire theory relies on the assumption that a steady state exists. However, during Stochastic Gradient Descent (SGD) training, the underlying electric field actually changes drastically at each step due to data being fed in mini-batches. It is questionable whether the concepts of steady state can still strictly hold in a constantly changing field.\n\nRegarding assumption: The paper assumes that high-level features behave like charged particles with uniform initial velocities. How sensitive is the model's performance to this assumption? Have the authors explored alternative physical analogies or initial conditions?", "limitations": "See weakness and questions", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strength:\nNovelty: The paper's main strength lies in its novel perspective of bridging concepts from physics (the Vlasov-Poisson system) with deep learning. This provides an interpretable model for understanding OOD phenomena, moving beyond purely statistical approaches.\n\nStrong Empirical Performance: The proposed CBD method achieves state-of-the-art results on multiple benchmarks for both OOD detection and OOD generalization. This demonstrates the practical effectiveness of the physics-inspired approach.\n\nUnified Approach: The framework addresses both OOD detection and generalization within a single conceptual model. The basin of attraction serves as a unifying concept for identifying outliers and for quantifying and improving model robustness.\nCompatibility: The proposed method is a modular addition to existing network architectures. The experiments show that it can be combined with other advanced OOD generalization techniques (like SAM and SAGM) to yield further performance improvements.\n\nWeakness:\n1.The paper's entire theoretical foundation is based on treating feature vectors as charged particles and describing their behavior with the Vlasov-Poisson system. However, the paper does not sufficiently justify how much deviation will this approximation induce and why using this specific model.\n\n2.The Vlasov-Poisson system is difficult to handle in high-dimensional spaces. The paper applies this model to the high-dimensional feature space generated by deep networks, but it does not explicitly address the challenges posed by the \"curse of dimensionality.\"\n\n3.The framework introduces several new and non-standard hyperparameters, such as the loss weights (α, β), the wavevector scaling factor (σ), and the wavenumber (N). Hyperparameter analysis is required.", "was_revised": true, "final_justification": "Thanks a lot for the rebuttal. However, given the workload for modifying this paper, the reviewer decides to maintain the original score. For the next submission, if the writing clarity could be improved, the reviewer will give the acceptance score.", "rebuttal": "We appreciate your constructive feedback and acknowledgment of the innovative aspects of our work. Our detailed responses are as follows.\n\n---\n\n> Q1: **Weaknesses-1:** The paper... treating feature vectors as charged particles... how much deviation will this approximation...\n\nWe appreciate your concerns about modeling feature vectors as charged particles under the Vlasov-Poisson system. You questioned whether mapping computational entities to physical quantities introduces bias. We identify 3 potential sources of bias from your feedback (please note if we missed any) and address each by demonstrating our model's alignment with neural network.\n\n- **Appropriateness of Analogizing High-Level Features to Charged Particles:** This analogy represents mathematical abstraction leveraging collective behavior of interacting entities, not literal physical equivalence. Neural network features share key properties with particle systems: distributed nature (features occupy high-dimensional spaces with continuous distributions, analogous to particle phase space), interaction patterns (features influence each other through attention mechanisms and nonlinear transformations), conservation properties (information flow follows conservation principles similar to physical laws) [1,2], and statistical behavior (both systems benefit from statistical mechanics when handling large ensembles).\n\n- **Reasonableness of Feature Evolution Driven by Self-Consistent Electric Fields:** \nThe attention mechanism captures relationships between tokens, analogous to self-consistent electric fields where electrons (high-level features) move under Coulomb forces from all other electrons. This analogy reveals our approach as a novel form of interaction: (i) query-key interactions represent forces, with attention weights as force magnitudes; (ii) attention implements a mean field approximation where each feature interacts with the average effect of all others; (iii) queries function as test charges probing the electrostatic field generated by keys (source charges); (iv) self-consistency emerges as attention depends on features distribution. Coulomb forces thus provide a universal **interaction kernel** in high-dimensional feature space, unconstrained by the physical properties of real charged particles.\n\n- **Lack of Mass in High-Level Features Compared to Physical Electrons:** \nAs we answered reviewer $\\text{\\textcolor{red}{MQNz}}$ in question $\\text{\\textcolor{blue}{Q10}}$, in neural networks, we can interpret \"mass\" as the \"inertia\" or \"stability\" of features, so features with higher mass resist deviation during learning or optimization processes, just as mass affects trajectories in physics.\n\nMoreover, selective borrowing from physics is standard practice in physics-inspired machine learning. For example, [2] omits specific force laws and particle properties, while [4] disregards charge characteristics. Our charged particle modeling does constitute a form of bias. **However, structured deviation are often beneficial in deep learning: CNNs assume spatial locality, Transformers assume positional ordering, and RNNs assume sequential dependencies.** Our physical constraints enhance transparency and interpretability compared to black-box alternatives. While we acknowledge the limitations of this modeling choice, we believe the resulting performance gains and interpretability benefits justify the trade-off in our application domain. We hope this addresses your concerns.\n\n---\n\n> Q2: **Weaknesses-2:** \"Curse of dimensionality\" problem.\n\nWe appreciate your insightful concern regarding the curse of dimensionality. You are absolutely correct that the Vlasov-Poisson system, being a system of partial differential equations, becomes computationally prohibitive in high-dimensional settings. We would like to address how our approach mitigates these challenges:\n\n- Rather than solving the full time-dependent system $F(\\mathbf{z}, \\mathbf{v}, t)$ and $\\phi(\\mathbf{z}, t)$, which would require computationally expensive ODE integration, we directly parameterize the steady-state distributions $F^* (\\mathbf{z}, \\mathbf{v})$ and potential $\\phi^* (\\mathbf{z})$ using neural networks (Eq. 4). While this introduces some approximation error, it dramatically reduces computational complexity.\n- Instead of computing the integral $\\nabla^{2} \\phi^* (\\mathbf{z}) = -\\frac{1}{\\epsilon_{0}} ( \\int F^* (\\mathbf{z}, \\mathbf{v}) \\mathrm{d} \\mathbf{v} - \\rho_ {\\mathrm{ion}} (\\mathbf{z}) )$ required by the original Vlasov-Poisson formulation, we employ the Poisson-Boltzmann approximation: $\\phi^* (\\mathbf{z}) = - \\frac{1}{\\epsilon_0} (e^{-\\phi^* (\\mathbf{z})} - \\rho_{\\text {ion }}(\\mathbf{z}))$, which eliminates expensive integral computations.\n- In traditional physics, solving the Vlasov-Poisson equation typically requires complete discretization of the phase space. In our neural network approach, we conveniently employ Monte Carlo methods for random sampling of the phase space. As shown in the last subplot of Figure 6, increasing batch size allows our stochastic sampling to gradually approximate full grid-based methods.\n- We operate directly on high-level features extracted by deep networks, which have been shown to naturally capture low-dimensional manifold structure within high-dimensional spaces [5], effectively reducing the intrinsic dimensionality of the problem.\n\nThough a full theoretical analysis of high-dimensional convergence is challenging, our experiments show that these designs effectively overcome the curse of dimensionality in practice.\n\n---\n\n> Q3: **Weaknesses-3:** Hyperparameter analysis.\n\nWe have already provided hyperparameter analysis for all mentioned parameters in the Experiments section: loss weights (α, β) are analyzed in the \"Hyperparameter Analysis\" subsection with results in Figure 6 (subplots 4-5), while the wavevector scaling factor (σ) and wavenumber (N) are analyzed in the \"Plane Wave Analysis\" subsection with results in Figure 7.\n\n---\n\n> Q4: **Questions-1:** Regarding In-Distribution Structure.\n\nWe believe there is a misunderstanding. Our method does **not** assume unimodal in-distribution data. The unimodal illustrations are purely for visualization clarity, just as commonly seen in other OOD detection or DA papers.\n\nOur theoretical framework imposes **no constraints** on the shape of the InD—it accommodates arbitrary multimodal distributions. The steady-state formulation characterizes the equilibrium of the underlying dynamical system, not the geometric structure of the data distribution itself.\n\n---\n\n> Q5: **Questions-2:** Regarding the Model Choice.\n\nThe Vlasov-Poisson system lies at the heart of our framework because it uniquely captures the collective interactions of high-level features as \"charged particles\": the **\"clustering\"** of features under InD data (i.e., convergence toward steady state) versus the **chaos** or **instability** of features under OOD data (see Figure 4). This arises from the system's kinetic-potential energy balance, enabling us to quantify distributional deviations through a physics-motivated lens, which providing novel theoretical insights for OOD problems.\n\nAlternative dynamical systems typically describe **individual** particle trajectories or employ low-order approximations, sacrificing the precise modeling of collective effects that is essential for comprehensive InD/OOD characterization. Energy-based models similarly rely on static energy minimization to identify InD stable states and OOD high-energy anomalies, but neglect the **collective** interactions that are critical for capturing temporal dependencies and nonlinear effects in phase space. We kindly refer you to our response to Reviewer $\\text{\\textcolor{red}{jZnV}}$'s $\\text{\\textcolor{blue}{Q1}}$, which provides detailed motivation for our modeling approach.\n\n---\n\n> Q6: **Questions-3:** Regarding steady state.\n\nThanks for raising concerns about the steady-state assumption, which lies at the heart of our theoretical framework. To address this, we offer the following clarifications:\n\n- With small batch sizes, the steady state is indeed more susceptible to disruption, as noted in the hyperparameter analysis within our Experiments section (see the final subplot in Figure 6). Our results demonstrate that larger batch sizes make the steady state increasingly robust, with stabilization occurring beyond a certain threshold.\n- In our experiments, the steady state solution exhibits residual fluctuations due to limited sampling from the batch sizes used, resulting in incomplete field descriptions per iteration. Through repeated sampling, the learning process gradually refines the field’s representation, requiring multiple iterations for an accurate steady-state solution.\n\n---\n\n> Q7: **Questions-4:** Regarding assumption.\n\nWe appreciate this question about our uniform initial velocity assumption. We conducted additional experiments with different initialization strategies: uniform vectors with values of 0.5 matching the feature dimension, and random vectors of matching dimension. Random initialization slightly underperforms uniform, but differences are minimal, showing robustness. The uniform assumption maintains effectiveness while simplifying computation.\n\n**Table 1: OOD Detection on CIFAR-100**  \n||||\n| - | :-: | :-: |\n| **Initial_Velocity** | **FPR95 ↓**  | **AUROC ↑** |\n| **1** | 34.81 | 90.90 |\n| **0.5** | 35.20 | 90.47 |\n| **Random** | 38.29 | 89.52 |\n||||\n\n\n**Table 2: OOD Generalization on PACS**  \n| | |\n| :- | :-: |\n| **Initial_Velocity**| **Accuracy ↑** |\n| **1** | 87.7 |\n| **0.5**  | 87.7 |\n|**Random**|87.4|\n||||\n\n---\n\n[1] Neural conservation laws: A divergence-free perspective. NeurIPS 2022.  \n[2] Hamiltonian neural networks. NeurIPS 2019.  \n[3] Mean-field and kinetic descriptions of neural differential equations. arXiv 2020.  \n[4] Poisson flow generative models. NeurIPS 2022.  \n[5] Representation learning: A review and new perspectives. TPAMI 2013."}, {"review_id": "aJKAYB9woy", "reviewer": "Reviewer_jZnV", "summary": "This paper introduces a novel physics-inspired framework called Collective Behavior Dynamics (CBD) to address out-of-distribution (OOD) detection and generalization. It conceptualizes high-level features as charged particles evolving under a self-consistent electric field, modeled via the Vlasov–Poisson system, and defines the “basin of attraction” as the region where in-distribution (InD) features converge to a steady state. The authors show that OOD samples can be identified as those falling outside this basin, while model generalization can be enhanced by enlarging the basin through dispersion relation regularization. According to evaluation results, CBD achieves competitive performance across multiple OOD detection and generalization benchmarks, and is compatible with existing methods for further performance gains. This approach bridges dynamical systems theory with deep learning, offering both theoretical insights and practical benefits for robust model behavior under distribution shifts.", "questions": "- What is the theoretical or empirical justification for modeling high-level features as charged particles governed by the Vlasov–Poisson system, as opposed to using conventional statistical methods? It is beneficial if the authors provide clearer rationale and comparative discussion that contrasts this physics-inspired approach with established statistical paradigms (e.g., energy-based models, density estimation, confidence scoring). What unique modeling capabilities or interpretability does the proposed framework offer?\n- Are there specific cases where CBD fails to detect OOD or generalize well? What are the limitations in terms of types of shifts or datasets?\n- Can the authors evaluate CBD in additional domains (e.g., time series, tabular data) or on more realistic OOD scenarios (e.g., long-tailed or adversarial settings)?", "limitations": "- The experimental design is limited to imaging data. Generalization beyond vision tasks is not discussed, leaving unclear how well the method transfers to domains like healthcare, finance, or natural language, where OOD robustness is also critical.\n- The proposed approach introduces multiple MLP branches (for learning potential and distribution functions), PDE-based losses, and Fourier-domain regularization, all of which significantly increase computational and memory overhead.\n- Several derivations (e.g., use of plane wave decomposition, homogeneous steady state) rely on simplifying assumptions (e.g., uniform ion background, constant velocities) that may not capture the full complexity of real-world feature distributions.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- Treating high-level features as particles in a dynamical system offers a new paradigm for OOD detection/generalization grounded in physical systems rather than statistical heuristics.\n- The method supports both OOD detection and generalization using the same underlying principles, and defines clear, interpretable loss functions (Poisson, Vlasov, and dispersion losses) tied directly to physical laws.\n- Code is provided for reproducibility purposes.\n\nWeaknesses:\n\n- This paper lacks a comprehensive comparison between physics-based and statistical approaches for OOD detection and generalization. While the proposed method is rooted in a novel physics-inspired framework, the authors do not provide sufficient discussion on how it fundamentally differs from or improves upon traditional statistical methods. Specifically, it remains unclear why modeling high-level features as particles under self-consistent fields offers a more effective or principled solution compared to well-established statistical techniques such as energy-based models, density estimation, or confidence-based scoring. A deeper theoretical and empirical analysis comparing the strengths, limitations, and use-case suitability of both perspectives would significantly strengthen the motivation and positioning of the proposed approach.\n- The framework assumes that high-level features behave like charged particles in a self-consistent field and reach a steady state. While this is an elegant abstraction, it lacks empirical or theoretical grounding to show that this analogy meaningfully holds for deep neural networks.\n- The limitation of the proposed theoretical framework is not discussed. In particular, this framework doesn’t address conditions under which the Vlasov–Poisson-based formulation might fail or be unstable, especially in highly non-linear or high-dimensional feature spaces.\n- The method is built around visual features extracted by CNNs or ViTs, and it's unclear whether the approach generalizes well to non-image modalities (e.g., text, graphs, time series).\n- The writing is overly complex and lacks clear organization, which makes it challenging to fully grasp the core concepts of CBD, particularly its application to OOD detection and generalization, as well as the accompanying theoretical analysis.", "was_revised": true, "final_justification": "As most of my concerns have been addressed, I have decided to increase my score to 4.", "rebuttal": "Thank you for your thoughtful suggestions and constructive feedback. We're delighted that you see our work as providing a fresh perspective for OOD detection and generalization. We have responded to your comments and questions below.\n\n---\n\n> Q1: **Weaknesses-1:** This paper lacks a comprehensive comparison between... how it fundamentally differs from...\n> **Weakness-2:** The framework assumes... this analogy meaningfully holds for deep neural networks.\n> **Questions-1:** What is the theoretical or empirical justification for modeling high-level features as charged particles... What unique modeling capabilities or interpretability does the proposed framework offer?\n\nWe appreciate your thoughtful question. Since the first and second point in Weaknesses and the first point in Questions address similar concerns, we respond to both together. The following we will explore the core aspects of this question from three aspects:\n\n**1. Why model features as charged particles under Vlasov–Poisson system?**\n\nOur adoption of the Vlasov-Poisson system is methodologically motivated by the need to explicitly model global interactions among high-level features in deep network latent spaces. The theoretical foundation is as follows:\n\n- Current some OOD detection and generalization approaches utilizing KNN methods (score computation based on sample-neighbor distances) [1] or local graph-based structures [2,3] (scores derived from graph isomorphism/connectivity) **implicitly** encode feature interactions. These interactions naturally parallel physical forces, where similar features exhibit mutual attraction and dissimilar features demonstrate repulsion, analogous to electrostatic interactions among charged particles. The Vlasov-Poisson system **explicitly** captures this phenomenon through a self-consistent field related formulation: each particle (representing high-level features) $\\mathbf{z}$ evolves under the collective field $E(\\mathbf{z}, t)$ generated by all particles (governed by equation $\\frac{\\partial F}{\\partial t}+\\mathbf{v} \\cdot \\nabla_\\mathbf{z} F+\\nabla_\\mathbf{v} \\cdot \\mathbf{E}=0$), effectively aggregating global information. The resulting steady-state particle distribution function $F^* (\\mathbf{z}, \\mathbf{v})$ and steady-state potential $\\phi^* (\\mathbf{z})$ serve as natural scoring mechanisms.\n- Crucially, while traditional approaches such as KNN or graph-based methods rely on **localized feature interactions** (such as $p(x)= {\\textstyle \\prod_{i}} p(x_i) $), the Vlasov-Poisson system enables **globally-informed** particle dynamics. This fundamental distinction provides theoretical justification for expecting superior OOD detection and generalization performance.\n- As elaborated in our response to reviewer $\\text{\\textcolor{red}{KKbs}}$'s question $\\text{\\textcolor{blue}{Q1}}$, the physical quantities emerging in the Vlasov-Poisson system (such as particles, self-consistent electric forces, and mass) all have corresponding abstract interpretations derived from the properties of high-level features.\n\n**2. Comparison to traditional statistical methods.**\n\n|||||||\n|-|-|-|-|-|-|\n| **Paradigm**| **Modeling Assumption**| **OOD Mechanism**| **Sample Interactions**| **Key Formula/Concept**| **Interpretability**|\n| **Energy-based Models (EBMs)** | Modeling with energy functions (logits or feature space energy)| Low energy → InD sample, high energy → OOD | [✖] No inter-sample/features interactions| $\\mathcal{E}(x)=−\\log ⁡p_\\theta (x)$, where $\\mathcal{E}$ is the energy of sample $x$| Intuitive in low-dimensional spaces, but hard to interpret globally|\n| **Density Estimation**| Models the probability density of data| Low probability density → OOD| [⚠] **Local** inter-sample interactions (via density estimation) | $p(x)= {\\textstyle \\prod_{i}} p(x_i) $, density function of feature vectorx $x$| Can capture local trends but lacks interpretability for high-dimensional spaces|\n| **Confidence Scoring**| Confidence score based on model output (e.g., softmax or ensembles)  | Low confidence → OOD| [✖] No inter-sample interactions| $C(x) = \\text{softmax}(f(x))$, confidence score from model output| Heuristic and intuitive, but limited by model's ability to generalize|\n| **Ours (Vlasov–Poisson System)** | Modeling feature embeddings as charged particles interacting via a self-consistent field | OOD samples cannot reach steady state | [✔] **Globally** explicit inter-features interactions| $\\frac{\\partial F}{\\partial t}+\\mathbf{v} \\cdot \\nabla_\\mathbf{z} F+\\nabla_\\mathbf{v} \\cdot \\mathbf{E}=0$, evolution equation for particle distribution, $\\mathbf{E}$ is the electric field| Physically grounded interpretation through particle interactions (e.g., attraction/repulsion/chaos) |\n|||||||\n\nA key distinction is that most conventional methods produce point-wise scores that fail to account for how samples relate structurally to one another, or only consider local neighborhood relationships. In contrast, the Vlasov-Poisson system inherently captures collective interactions, making it particularly well-suited for subtle OOD problems, as we demonstrate through the **sharp gradient** properties near decision boundaries established in Corollary 2.3 of our paper.\n\n**3. Unique modeling advantages & interpretability.**\n\n- From an interpretability perspective, the physical metaphor (e.g., attraction/repulsion/chaos) provides an **intuitive lens** for understanding why certain samples are classified as OOD—not simply due to \"low confidence\" or \"low density\" scores, but because they fundamentally violate the governing field structure learned from InD data.\n- A key methodological distinction is that our method constitutes a system governed by PDEs that models the \"collective dynamics\" of the entire distribution. Unlike conventional static approaches, each sample is assessed within its relational context to all other samples through self-consistent field interactions, enabling us to capture global distributional shifts and inter-sample dependencies critical for robust OOD problems.\n\n---\n\n> Q2: **Weaknesses-3:** The limitation of the proposed theoretical framework is not discussed. In particular... **Questions-2:** Are there specific cases where CBD fails to detect OOD or generalize...\n\nYou raise a valid concern. Our hyperparameter analysis (final subplot in Figure 6) demonstrates that insufficient batch sizes during training can indeed degrade performance, we provide additional details in our response to reviewer $\\text{\\textcolor{red}{KKbs}}$'s question $\\text{\\textcolor{blue}{Q6}}$. Regarding the dimensionality issues you mention, we employ several mitigation strategies, which we address comprehensively in our response to Reviewer $\\text{\\textcolor{red}{KKbs}}$'s question $\\text{\\textcolor{blue}{Q2}}$.\n\n---\n\n> Q3: **Weaknesses-4:** The method is built around visual features... non-image modalities... **Questions-2:** Can the authors evaluate CBD in additional domains... **Limitations-1:** The experimental design is limited to imaging data...\n\nThank you for raising this point. We respectfully encourage you to refer to $\\text{\\textcolor{blue}{Q3}}$ in our response to reviewer $\\text{\\textcolor{red}{Sdub}}$.\n\n---\n\n> Q4: **Weaknesses-5:** The writing is overly complex and lacks clear organization...\n\nThank you for highlighting the writing issues. Reviewer $\\text{\\textcolor{red}{MQNz}}$ raised similar concerns in $\\text{\\textcolor{blue}{Q1 $\\sim$ Q7}}$ and $\\text{\\textcolor{blue}{Q9}}$, which we've addressed in detail. We encourage you to refer to those responses. We will integrate these changes into the revised manuscript to enhance readability.\n\n---\n\n> Q5: **Limitations-2:** ... increase computational and memory overhead.\n\nWe acknowledge that our physics-inspired approach incurs additional computational overhead. To mitigate this, we employed two techniques to reduce this overhead, including directly parameterizing $F^* (\\mathbf{z}, \\mathbf{v})$ and $\\phi^* (\\mathbf{z})$ rather than the full time-dependent $F(\\mathbf{z}, \\mathbf{v}, t)$ and $\\phi(\\mathbf{z}, t)$, and using the Poisson-Boltzmann approximation $\\phi^* (\\mathbf{z}) = - \\frac{1}{\\epsilon_0} (e^{-\\phi^* (\\mathbf{z})} - \\rho_{\\text {ion }}(\\mathbf{z}))$ in place of integral $\\nabla^{2} \\phi^* (\\mathbf{z}) = -\\frac{1}{\\epsilon_{0}} ( \\int F^* (\\mathbf{z}, \\mathbf{v}) \\mathrm{d} \\mathbf{v} - \\rho_ {\\mathrm{ion}} (\\mathbf{z}) )$. The primary computational cost arises during training, driven by GPU memory demands for storing intermediate computation graphs and performing automatic gradient calculations. During inference, however, these intermediate computations are unnecessary, resulting in minimal impact on inference speed. Moreover, the added MLP branches contribute only a small fraction of model parameters (2.7% for ViT-b16 and 4.7% for ResNet-18, as detailed in Appendix Table 5), ensuring negligible effects on model storage and inference costs.\n\n---\n\n> Q6: **Limitations-3:** Several derivations... may not capture the full complexity of real-world feature distributions.\n\nThank you for raising this point. We believe there may be a misunderstanding here. \n\n- For OOD detection, our post-hoc method CDB-De does not modify pre-trained parameters, so InD feature distributions remain unaffected by physical assumptions. \n- For OOD generalization, our physical loss serves as a strong structural constraint. Rather than merely acting as regularization, this constraint actively guides the feature learning process to discover representations that respect underlying physical laws, thereby improving robustness across domains. \n\nWe acknowledge these potential limitations and plan to explore relaxed assumptions in future work.\n\n---\n\n[1] Out-of-distribution detection with deep nearest neighbors. ICML 2022.  \n[2] Environment-aware dynamic graph learning for out-of-distribution generalization. NeurIPS 2023.  \n[3] Learning invariant graph representations for out-of-distribution generalization. NeurIPS 2022."}, {"review_id": "cYKPfSDK9E", "reviewer": "Reviewer_Sdub", "summary": "This paper introduces a novel, physics-inspired framework CBD to address Out-of-Distribution (OOD) detection and generalization. The authors conceptualize high-level features extracted from neural networks as interacting charged particles governed by the Vlasov-Poisson equations, analyzing their collective behavior within a dynamically evolving self-consistent electric field. They formally define a basin of attraction that distinguishes In-Distribution (InD) from OOD data and propose two distinct but complementary methods derived from this framework: (i) an OOD detection method based on identifying steady-state solutions to distinguish InD and OOD features, and (ii) an OOD generalization method leveraging a dispersion relation to stabilize perturbations and enlarge the basin boundary.", "questions": "Please refer to Weaknesses.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The paper uniquely leverages theoretical constructs from physics (Vlasov-Poisson dynamics, charged particle behavior) to fundamentally rethink the OOD detection and generalization problems, offering a refreshing theoretical perspective distinct from prevailing statistical and uncertainty-based methods.\n2. The theoretical underpinning is compelling, outlining why the proposed dynamics-based approach naturally differentiates between InD and OOD data through steady-state solutions and basin boundaries.\n3. The method can be integrated smoothly with other advanced OOD generalization techniques, adding to its practical utility and adaptability.\n\nWeaknesses:\n1. The method requires the simultaneous estimation of complex PDE solutions, potentially resulting in increased GPU memory usage and reduced computational efficiency, especially in very large-scale or resource-constrained environments.\n2. While some hyperparameter analyses are presented, a deeper understanding of the sensitivity and stability related to PDE discretization parameters, regularization strengths, or solver accuracy would significantly enhance the method’s clarity and reproducibility.\n3. The experimental evaluation is limited to relatively standard feature-extraction architectures, specifically ResNet and ViT, leaving the effectiveness of the approach on more diverse network architectures or modalities (e.g., text or multimodal data) unexplored.", "was_revised": true, "final_justification": "The authors' responses address the majority of my concerns. I am therefore willing to raise my overall evaluation score.", "rebuttal": "Dear reviewer Sdub, thank you for your thoughtful review and constructive feedback on our manuscript. We are pleased that you recognize the refreshing theoretical perspective our proposed method brings, which distinguishes it from conventional approaches in the field. We address your specific comments and suggestions in detail below:\n\n---\n\n> Q1: **Weaknesses-1:** The method requires the simultaneous estimation of complex PDE solutions, potentially resulting in increased GPU memory usage and reduced computational efficiency, especially in very large-scale or resource-constrained environments.\n\nThank you for your comment. A similar concern was raised by reviewer $\\text{\\textcolor{red}{jZnV}}$ in $\\text{\\textcolor{blue}{Q5}}$, and we have provided a detailed response there. Briefly, we acknowledge that our method introduces additional computational overhead during training due to the storage of intermediate computation graphs and automatic gradient calculations. To mitigate this, we adopt two strategies: \n- Directly parameterizing steady-state solution instead of the full time-dependent functions.\n- Using the Poisson-Boltzmann approximation to avoid solving the full integral form of the PDE. \n\nThese design choices reduce GPU memory usage while preserving the core physics-informed structure. At inference time, the overhead is negligible, and the additional model parameters account for only a small fraction of the total. Please refer to our response to reviewer $\\text{\\textcolor{red}{jZnV}}$'s $\\text{\\textcolor{blue}{Q5}}$ for further details.\n\n---\n\n> Q2: **Weaknesses-2:** While some hyperparameter analyses are presented, a deeper understanding of the sensitivity and stability related to PDE discretization parameters, regularization strengths, or solver accuracy would significantly enhance the method's clarity and reproducibility.\n\nWe appreciate this insightful suggestion. However, we'd like to clarify a potential misunderstanding: Our method does **not** rely on an ODE/PDE solver to iteratively fit the particle distribution function $F(\\mathbf{z}, \\mathbf{v}, t)$ and electric potential $ \\phi (\\mathbf{z}, t)$ over time steps. Instead, we use MLPs to directly predict the steady-state \\$F^* (\\mathbf{z}, \\mathbf{v})$ and \\$\\phi^* (\\mathbf{z})$ in a single forward pass, completely bypassing numerical time integration. This fundamental distinction means that conventional ODE solver parameters (discretization step sizes, solver tolerance, numerical stability constraints, etc.) are not relevant to our method.\n\nThat said, we recognize the value in analyzing the sensitivity of MLPs hyperparameters (e.g., layer depth, width), which could impact the stability of the steady-state approximation. We have examined these variations (hyperparameter tuning details are in our response to reviewer $\\text{\\textcolor{red}{MQNz}}$'s question $\\text{\\textcolor{blue}{Q12}}$). Below, we present the sensitivity analysis results for several relevant parameters:\n\n**Table 3: OOD Detection on CIFAR-100 with MLP Hidden Dimension of 512**\n|  |  |  |\n| :--- | :---: | :---: |\n| **Layer_depth** | **FPR95 ↓** | **AUROC ↑** |\n| **1** | 39.30 | 87.14 |\n| **2** | 34.81 | 90.90 |\n| | | |\n\n\n\n**Table 4: OOD Detection on CIFAR-100 with MLP Depth of 2 Layers**\n|  |  |  |\n| :--- | :---: | :---: |\n| **Hidden_dim** | **FPR95 ↓** | **AUROC ↑** |\n| **128** | 40.35 | 86.93 |\n| **256** | 38.52 | 88.20 |\n| **512** | 34.81 | 90.90 |\n| **1024** | 35.25 | 90.83 |\n|  |  |  |\n\nTable 3 and Table 4 show that performance scales with the number of MLP parameters, underscoring the key role of these hyperparameters in improving the model's ability to approximate steady state solutions. While increasing MLP architectural complexity delivers gains, returns diminish beyond a certain threshold (e.g., hidden dim=512 vs. 1024). This highlights the need to balance MLP intricacy with computational efficiency during tuning. We believe these insights will inform further optimizations of our approach for real-world applications.\n\n---\n\n> Q3: **Weaknesses-3:** The experimental evaluation is limited to relatively standard feature-extraction architectures, specifically ResNet and ViT, leaving the effectiveness of the approach on more diverse network architectures or modalities (e.g., text or multimodal data) unexplored.\n\nWe agree that more experimental evaluation would be useful to understand the details ofinteraction andenhancement. We would like to address this concern from several perspectives:\n\n- Our experimental design strictly **align with** established evaluation protocols for OOD detection and generalization in the visual domain. Specifically, we adopted the experimental **setups from the well-regarded benchmark frameworks OpenOOD [1] and DomainBed [2]**, including their datasets, evaluation metrics, and procedures. This ensures our results are comparable and consistent with recent existing works [3,4,5]. The datasets and backbone network models were carefully selected for their broad representativeness and ability to thoroughly validate the effectiveness of our approach.\n\n- Extending experimental evaluation from visual data to text, tabular data or multimodal data would require fundamental redesign of our method and development of specialized components for different modality characteristics. Text and multimodal OOD problems differ inherently from visual OOD problems and require distinct technical approaches, including but not limited to: prompt engineering, modality-specific fine-tuning strategies, cross-modal alignment techniques, and multimodal fusion mechanisms. **These technical challenges constitute separate research directions in their own right**.\n\n- Nevertheless, to address your suggestion and preliminarily explore the extensibility of our method, we conducted exploratory experiments on multimodal model CLIP. Specifically, we followed the ERM with CLIP method described in [6] and adopted the following experimental setup: we first initialized the classifier head weights using text embeddings extracted by the CLIP text encoder. Then, we fine-tuned the CLIP image encoder along with two additional MLP branches, with the overall experimental configuration following the standard implementation from [7]. Unlike [7] which only used standard ERM loss, we additionally integrated our three proposed loss components during training: the Vlasov loss $\\mathcal{L}_ {\\text{Vlasov}}$, the Poisson loss $\\mathcal{L}_ {\\text{Poisson}}$, and the Dispersion Relation loss $\\mathcal{L}_ {\\text{disp}}$. The results are shown in Table 5. This preliminary multimodal experiment validates the potential extensibility of our approach. We hope this additional multimodal experiment and the detailed explanation above effectively address your concerns while clearly indicating that our current research remains focused on image-based OOD problems and their theoretical foundations.\n\n\n**Table 5: Accuracy on the PACS and OfficeHome Datasets with Domain Shift.**\n\n|||||\n| :--- | :---: | :---: | :---: |\n| **Method** | **Backbone** | **PACS** | **OfficeHome**|\n| **Zero-shot** | CLIP |96.2 | 82.0 |\n| **ERM** | CLIP | 96.1 | 83.3|\n| **ERM (with our loss)**| CLIP | 96.4 (${\\small \\text{\\textcolor{red}{+0.3}}}$) | 84.0 (${\\small \\text{\\textcolor{red}{+0.7}}}$) |\n|||||\n\nWe fully agree with your suggestion. For the specific cases you mentioned, our method will require further customization. The approach based on discrete electric fields is conceptually similar to addressing OOD and related issues, and we plan to further explore this direction in future work.\n\n---\n[1] Openood: Benchmarking generalized out-of-distribution detection. NeurIPS 2022.  \n[2] In Search of Lost Domain Generalization. ICLR 2021.  \n[3] Gradient-Guided Annealing for Domain Generalization. CVPR 2025.  \n[4] Extremely Simple Activation Shaping for Out-of-Distribution Detection. ICLR 2023.  \n[5] Out-of-distribution detection based on in-distribution data patterns memorization with modern hopfield energy. ICLR 2023.  \n[6] Clipood: Generalizing clip to out-of-distributions. ICML 2023.  \n[7] Robust fine-tuning of zero-shot models. CVPR 2022."}]

Author rebuttal:

Meta-review: The paper models features as charged particles under a self-consistent field to unify OOD detection and generalization. It achieves strong benchmark results and complements existing methods, and the authors’ thorough rebuttal further demonstrates its value. Therefore, I recommend acceptance.
Decision: Accept (poster)