Interaction Field Matching: Overcoming Limitations of Electrostatic Models
Authors: S. I. Manukhov
Alexander Kolesov
V.V. Palyulin
Alexander Korotin
Venue: ICLR 2026 Poster (2026)
Keywords: generative models
distribution transfer
electrostatics
Primary area: generative models

Abstract: Electrostatic field matching (EFM) has recently appeared as a novel physics-inspired paradigm for data generation and transfer using the idea of an electric capacitor. However, it requires modeling electrostatic fields using neural networks, which is non-trivial because of the necessity to take into account the complex field outside the capacitor plates. In this paper, we propose Interaction Field Matching (IFM), a generalization of EFM which allows using general interaction fields beyond the electrostatic one. Furthermore, inspired by strong interactions between quarks and antiquarks in physics, we design a particular interaction field realization which solves the problems which arise when modeling electrostatic fields in EFM. We show the performance on a series of toy and image data transfer problems.

Published as a conference paper at ICLR 2026
INTERACTION FIELD MATCHING: OVERCOMING LIMITATIONS OF ELECTROSTATIC MODELS
Stepan Manukhov
Alexander Kolesov*
MSU Faculty of Physics, Moscow, Russia Applied AI Institute, Moscow, Russia manukhov2000akk gma 11.com
Applied Al Institute, Moscow, Russia AXXX, Moscow, Russia a alexander www.sndedr...leeeeemmmama. com
Vladimir V. Palyulin
Alexander Korotín
Applied Al Institute, Moscow, Russia v. v.palyuling gmail com.
Applied AI Institute, Moscow, Russia AXXX, Moscow, Russia amaa eexkorotin@gma com
ABSTRACT
Electrostatic field matching (EFM) has recently appeared as a novel physics- inspired paradigm for data generation and transfer using the idea of electric capacitor. However, it requires modeling electrostatic fields using neural networks, which is non-trivial because of the necessity to take into account the complex field outside the capacitor plates. In this paper, we propose Interaction Field EFM which allows using general interaction
Matching (IFM), generalization fields beyond the electrostatic one. Furthermore, inspired by strong interactions between quarks and antiquarks in physics, we design a particular interaction field realization which solves the problems which arise when modeling electrostatic fields in EFM. We show the performance on series of toy and image data transfer problems, Our code is available https::////. / com/ justkolesov/ InteractionE ie MeM.anneieeaeaMMcciccccccc
E
E(X)
L
2
PP PP(xg)
Q(xa)
Electrostatic Field Matching Kolesov et al. 2025 EFM).
(b) Interaction Field Matching
(IFM, ours).
Figure 1: Electrostatic Field Matching Kolesov et al. 2025 EFM) and our Interaction Field
Matching (IFM) concepts. Two D-dimensional distributions P(-), Q(-) are placed in D+1 at z and = L (a) In EFM, the distributions are interpreted as charges creating capacitor-like electric field. Movement along these field lines transfers the distributions, but requires consideration of all directions of the field lines. (b) Our IFM is generalization of the EFM arbitrary interactions between charges. One possible realization IFM motivated by the strong interaction between quarks. This realization does not have backward-orientee lines and has smaller curvature of lines.
"Equal contribution.
Moscow State University

---

Published as a conference paper at ICLR 2026
INTRODUCTION
While diffusion (Sohl-Dickstein et al. 2015 Ho et al. 2020) and flow matching (Liu et al. 2023
Lipman al., | Albergo Vanden- Eiinden (023) models dominate current research in deep generative modeling, new paradigm grounded in Coulomb electrostatics has emerged (Xu et al.
2022; Kolesov et al. 2025 Cao & Zhao 2024, Cao al. 2024, Xu al. 2023). Early work this direction introduced Poisson Flow Generative Models (Xu et al. 2022 2023 PFGM), focusing noise-to-data generation. More recently, Electrostatic Field Matching (ooeesvv et al. 2025 EFM)
generalized this framework, enabling electrostatic models to solve data-to-data transfer problems.
Electrostatic Field Matching (EFM) draws inspiration from electric capacitors, modeling input and target distributions positive and negative electrostatic charges, respectively. The method performs distribution transfer by following electrostatic field lines (Fig. Ta). While conceptually simple, EFM faces significant practical challenges: it requires accounting for all field iines-including backward- oriented ones (dotted lines in Fig. la which exhibit high curvature and span the entire space. This makes them difficult to model, as the necessary training volume becomes unbounded.
In this paper, we tackle the limitations of EFM 2.3 and deliver the following main contributions:
Theory. propose Interaction Field Matching (IFM), a generic paradigm for distribution transfer rooted in pairwise interactions between particles from input and target distributions ($3.3
3.5). Compared to EFM that relies on the electrostatic field, our approach allows us to leverage general interaction fields (beyond the Coloumb electrostatics) that satisfy certain physics-inspired properties such as the flux conservation and the generalized superposition principle (33.2)
Methodology practice. Inspired by the strong interaction of quarks and antiquarks in physics (3.1)) design particular realization of the interaction field (8f.4) which has several preferable properties compared to the electrostatic field: (a) the field lines have almost straight segments, (b) the field vanishes outside the between particles and (c) it allows using the Minibatch Optimal Transport Pooladian et al. 2023) to enforce certain properties on the transfer map.
We showcase the performance of IFM on a series of toy and image data transfer problems ($4)
2 BACKGROUND AND RELATED WORKS
In this section, we first recall the concepts of the basic high- -ich-diinnsionall electrostatic ($2.1). Then we discuss its application to generative modeling and data transfer problems using the example of EFM ($2.2. Finally, in 22.3 discuss the limitations of EFM which motivated our study.
2.1 ELECTROSTATICS
We recall the fundamental principles of electrostatics necessary for understanding electrostatic- generative models. A detailed treatment of three-dimensional electrostatics can be found any standard electricity and magnetism textbook, e.g., (Landau Lifshitz 1971. Chapter 5). The generalization of electrostatics to high-dimensionall spaces is discussed in (Caruso et al. 2023).
The electrostatic field. Let : DDD
TRD. The distribution may contain both positive and negative charges and is assumed to have finite total charge (J [q(x))xxx dx o0). point I it produces the electrostatic field E RDD + RD:
be the density of charge distribution
E(x) = /
x') Sp ||x x' x'11
1 (x-x')
9 (x')dxxx
(I)
where Sp-1 is the surface area of an (D- (10-1)-cinng -immenniional sphere with unit radius. That is, the field a weighted sum of fields from all charges x', where closer charges yield stronger field.
Electric field strength lines. An electric field strength line is a curve x(7) E IR,, T [a, b] CR whose tangent to each point is parallel to the electric field that point. In other words:
dx(r))
E(x).
(2)
Electric field lines are a key concept for electrostatic generative models such as PFGM and EFM.
2

---

Published as a conference paper at ICLR 2026
2.2 ELECTROSTATIC FIELD MATCHING (EFM)
The first application of electrostatics to generative modeling problems was carried out in the works of (Xu et al. 2022 2023 PFGM), where the authors proposed model applicable noise-to-data generative problems. Electrostatic Field Matching (EFM) extends the application of electrostatics to the case of data-to-data transfer, and uses previously unconsidered properties of electric field lines. We describe here EFM since it is more general than PFGM, and our work is built upon it.
EFM works with two data distributions P(X") and ), E RR The first distribution is assigned a positive charge, while the second distribution is assigned a negative charge. The distributions are placed in the extended space RP+ on the planes
and 2 L. respectively (see Fig. Ia). (D + ---dimensional capacitor. A point in this space has the form
One can think of it (x1, 32, DD 2) = (x, 2) = x E R P+1 The field is found from the superposition principle:
E(x) E
where E+ (x) and E_ (x) are the fields created by PP and Q(X ) respectively.
Then, as proved in the original paper, the movement along the field lines dx = E(X) performs the transfer between the distributions Px(*) and Q()), ). This fact has opened the possibility to use electrostatics both in data generation and transfer problems. Indeed, to move between data distributions, it is sufficient follow the electric field lines.
To obtain a distribution transfer model, one trains a neural network fa(-) : PR1 + RD to recover the normalized electric field
B(x) [EX] e.g., by using loss function
E(x)
Exl)fo(x)
1l2 min.
(4)
[E
Here, E(x) is calculated with (3 where E+ (x) approximated by empirical samples of TP(xX and Q( ), i.e., data. Monte Carlo averaging Eg is done on the points x around the plates. This set of
points if called the training volume; its selection is crucial but highly non-trivial (Xu al. 2022)
2.3 LIMITATIONS OF EFM
Despite its performance, EFM has a few weak spots coming from the properties of electrostatic fields:
Backward-orienter field lines. Each plate produces two sets of electric field lines (Fig. Tap. The first set ffrward-orienter lines) is directed toward the second plate. The second set (backward oriented lines) starts from the first plate in the opposite direction. practice, the forward-orienter
x+ ~ P(x+)
X² - P(Xg)
Q
y T(x))
T(x)))
Olxa)
0
-2
44
-4
-6
15
15
10
10
1
1
3
6
-15
7
-15
(a) EFM
(b) Our IFM (independent plan)
Figure 2: Limitations of the EFM & comparison with IFM. (a) The toy experiment (1 + 2 Gaussians) shows that even some cewwrrd-oriettel field lines can leave L. These trajectories have increased length and curvature. Moreover, the transfer along only the forward-oriented lines does not cover the target distribution (green point cloud does not coincide with the red one).
(b) Our realization of IFM ($3.4) does not have the above mentioned problems: the field lines between the planes are almost. straight, they do not extend beyond 2 >> and are enough to cover the entire target distribution.
3

---

Published as a conference paper at ICLR 2026
set of lines is chosen because requires less training volume and because these lines are less curved than the lines of the backward-oriente series. However, backward -rrcward lines play a critical role for the full coverage of the target distribution. The use of only to fully cover the distribution of Q(+), see the illustration Fig 2a
lines is not sufficient
2. Line termination problem. Even some forward-oriented field lines can pass the boundary 2 = L before reaching the second distribution. In such a case, the field line enters the region 2 L (see Fig. 2a) and requires further integration to come back to target distribution at 2 L. This problem complicates the data transfer procedure. Indeed, one has to design some criterion to decide whether the line terminates at L should be integrated further.
3. Training volume selection. From the first two problems follows the challenge of choosing the training volume, i.c., points x in equation for learning the field. For the correct transport between I-() and QQ-), it necessary to know not only the field between the plates (0 < L), but also beyond the plates (z > for lines leaving the boundary and & < for lines). Therefore, it is necessary choose large training volume for learning of the neural network. Moreover, the size of the required volume is initially unknown.
Below propose generalization of EFM which aims ease the above-mentioned problems.
3 INTERACTION FIELD MATCHING (IFM)
This section describes our proposed Interaction Field Mathcing (IFM), the generalization of the electrostatic paradigm in generative models. In $3.1 start by motivating the IFM with the strong interaction between subnuclear particles (quarks) in physics. In 33.2 we present the necessary requirements for interaction field required for our ideas to work. In R3.3] we formulate the main theorem devoted to transfer of distributions into each other by means of proper interaction field. The $3.4 describes particular realization of the field inspired by strong interactions. In $3.5 report the learning and inference algorithms. The proofs are in Appendix A.1
3.1 MOTIVATION: STRONG INTERACTION PHYSICS
q
To address EFM challenges, we propose utiliz- ing the strong interaction (Quevedo & Schachner 2024| 37.4) fundamental force binding subnu- clear particles. The smallest particles involved in this interaction called quarks.
q
typical configuration of the strong field shown in Fig. 3 highlighting key contrasts with electro- magnetic fields. At small distances, quark & and antiquark interact similarly to charged particles 4t- However, as the separation increases, the strong field lines become considerably straighter.
q
q Figure 3: Comparison of electrostatic interac- tion between charges qth (left) and strong in- teraction between quarks q, q (right). At small distances, the strong interaction resembles the electromagnetic interaction, but as quarks sepa- rate, the field lines straighten into string.
Unfortunately, strong field strength calculation re- quires complex quantum- connttu--eccanicaa computations. Although our work is motivated by quark interac- tions, unlike EFM and PFGM, our setting uses mod- ified physical interactions.
3.2 PROPERTIES PROPER INTERACTION FIELDS
Here list the most general requirements for the interaction field E(x) which are sufficient to perform data transfer. These requirements allow for broad ficxibility in the field design. In particular, could be electrostatic field (see Example bellow) Nevertheless, to preserve the concept of strong interaction, we will still refer to particles as quarks and antiquarks.
below).
Suppose that quark is located at the point X, R++ and an antiquark q the point X. R D+1 and produce interation field E(x) = E(x) nxx,, where n(x) is the unit vector tangent to the field line and (x) is the magnitude. We require the following properties of the interaction field E(x):
4

---

Published as a conference paper at ICLR 2026
1. The start and the termination of lines at (anti)quarks. For qq-pait with equal charges, the interaction field line must start at the quark and end at the antiquark:
dR(r) n (x((T))
(5)
x(T.) Xg. x(Ty)
Xg.
dS
where Ta, Ty correspond the initial and final points of the field line.
2. Flux conservation. For q4- qq-pair with equal charges, an interaction field must maintain the following property along the stream tube
E
E(x) dS const,
(6)
where dS is vector with length equal the area dS of the surface,
Figure 4: An illustration of the flux conservation.
where the considered stream tube rests. The direction of the vector is
orthogonal to the surface. In turn, E dS EdS eos = Ej dSi +
+ EDdSp denotes the inner product between the vectors E and ds. Informally, this property means that the number of field lines along the stream surface is constant, see Fig. We additionally assume that the total flux between quark-antiquart pair is proportional to the charge of the quark pair.
that creates this field, and does not depend on the relative position of the
3. Generalized superposition principle w.r.t. a transport plan. Consider two continuous distribu- tions q()), (7-) of quarks and antiquarks, respectively, and assume they have the same total charge. < ) be transport plan between these distributions, i.e., it satisfies the non-negativity property I (xxx> Xg) 2 0 and the marginal constraints I Txxg: Xg kg)xxg = g ((x)) Txxy Xg)dx, = 7(x)). Let Ex,,x, x. (x) denote the field produced by a pair of unit quark and unit antiquark located at Xq. Xq.
Then the field the system quarks and antiquarks q w.r.t. x is given by:
Eg (x) = ff Txx,, Xxy (X)
(x)
Physically, it means that E the average of fields of interacting pairs (xy. xg), and π (x2 x²)) characterizes the strength of pairing between quarks X, and Xg. For completeness, we note that the superposition principle can analogously stated for the discrete systems of quarks.
Properties 1-2 are formulated for quark-antiquark pair. However, if these properties are true for the qÿ-pair, they remain valid for more complex discrete and continuous systems.
Lemma 3.1 (On the field lines). Let qe) and 7(-) be two compactly supported (discrete or continuous) distributions of quarks and antiquarks. Let them satisfy J q(x)dx, = ((xq)xx,)))) Let the field of the quark-antiquark pair Ex,,x, (X) start at x, and terminate at X4 (Property 1), and conserve flux along the current tube (Property 2). Then the total field 7 from all quarks and antiquarks
(a) Start at (FPp(P) and end at supp(Q). except perhaps for the number of lines of zero flux.
(b) Conserve flux along the current tubes.
Example 3.2 (The electrostatic field). The electrostatic field (3) is special case of the interaction field satisfying properties 1-3 for an arbitrary transport plan. Indeed, property corresponds to the dipole field, property 2 follows from Gauss's theorem (Kolesov al. 2025 52.1), and property 3 is easy to check:
/ (x, + E.g (x,
(x)
dx
8
/ (x, )
Ex
=
where (x, is the electric field point x produced by point charge q +1 located at point x= Electrostatic fields are independent of the transport plan ग. i.e., changing the plan does not alter the field the total field of two charges itself is sum of two separate fields by these charges.
3.3 MAIN THEOREM
Let PP(x)) and Q(x)) be two D-dimensional data distributions. Similarly to EFM, we put these distributions in the extended space D+1 on the planes 2 = and 2 = L, respec- tively, see Fig. [b] Now assume that and are the distributions of quarks q and anti- quarks 7. respectively. Fix a transport plan π between them, e.g., set the independent one = == P x Q. Let E,, (x) be proper interaction field, i.e., satisfying properties 1-3 of 3.2
5

---

Published as a conference paper at ICLR 2026
For transport between the distributions, define map T (P) + supp(Q) that moves along the field lines by integrat- ing dx = dT where E(x) is defined by ( Field lines starting on the distribution PP(x)) can emanate in two directions: forward- oriented, directed toward (xx)). and backward-oriented, pointed initially in the opposite direction (see Fig. [5). The choice be- tween these directions of motion must be made stochastically. definition of the stochastic map T is provided in Appendix A.2
T
BBB
X
!
X
Then, for this map T(x,), we prove the following key theorem:
2=0 E
E
Theorem 3.3 (Interaction Field Matching).
Ler Ler P(x.) and Qxxq) be
two continuous data distributions that have compact support. Let x
Figure 5: Illustration of forward, backward lines.
be a random variable distributed as P(x,). Then Xg Txx,) is a random variable distributed as Q(xq):
xy P(x,) T(x;) xg Q(xq).
(9)
In other words, the movement along interaction field lines provably transfers P(x;) to Q(xg). The proof of the theorem is given in Appendix A.3
3.4
REALIZATION OF THE INTERACTION FIELD
0
X
X
e
Cz
d
L-d
L
0 d
L d
2
(a) Symmetric case.
(b) Shifted case.
Figure 6: Realization of the field between two quarks. In the range E [0, df and x [L d, L] the field lines curve toward the quarks, in the middle range [d, L the field lines straight. The string has an effective width op. beyond which the field value exponentially decreases. (a) Symmetric case. (b) The shifted case is obtained by proportional shift along the plane L
We present an interaction field realization that meets Properties 1-3 (33.2). motivated by quark interactions ( 3.I.I. This design eliminates backward-orientee lines and prevents field lines from going to z (Fig. (b). A schematic shown in Fig. while the detailed algorithm for calculating the field is formulated in Appendix A.4
Theorem 3.4 (Properties of our interaction field), Our realization of the interaction field E(x) satisfies the fundamental Properties 1-3 in $3.2 with additional characteristics:
Field lines never extend beyond & > L.
No tackward--rienteed lines exist.
The combination of these properties saves a given interaction field from EFM problems ($2.3) We give proof and provide additional discussions in App. A.5
3.5 LEARNING AND INFERENCE ALGORITIMM
To move between data distributions, it is sufficient to follow the interaction field lines. The lines can be found from the trained neural net approximating the interaction field E(x).
Training, To recover the interaction field E(-) in (D + 1)-dimensional points between the hyper-
planes, similarly to EFM, approximate with neural network fo(.) RRD+1 - RD+1 To begin

---

Published as a conference paper at ICLR 2026
with, need to define the training volume, i.e., the procedure to sample x for training. We use the following sampling scheme:
= (1 L
L n +(2),
(10)
where (Xz² Xg) ~ is sampled from the plan, 2
r(2) is the schedule distribution on [0. L] and [(2) is
Algorithm IFM Training
the amount of noise injected into the linear interpola-
Input: Distributions accessible by samples:
tion of x and K,, at level z. This scheme is inspired by
((xx))(z)) and ((x))((z L);
related works Kolesov et al. (2025); Xu et al.] (2022).
Transport plan Tx,;, xx) ) R x R - R;
NN approximator fo(-) :R P-11 R D+!
In generation experiments ($4 (4..2. we use uniform
distribution for r(z) and set {(2) equal to zero, be-
Output: The Iearned interaction field fo(-) Repeat until converged;
cause the linear interpolation of data point x, and
Sample [ batch (X.. X,) ~ xxp. X₂)
a noise sample xg already includes noise. In trans- lation experiments ($4.3). we use also uniform r(2) and (0, (z)) for E(z), where the variance o² is = 7 L z|. This formulation ensures that ((2) = 0 at 2 = 0 and = I., with the maximum 2- Also, it is worth
Sample [ coordinates ~ r(z):
Compute noise ?(=) as car (see App
Calculate batch X (1 i )),, + XG
Calculate Egg (X) for all pairs following
Calculate E(X) with 7
noise occurring at the middle noticing that (1O) is just one of many possible ways to define intermediate points between the plates, and our method does not have direct connection to the train- ing procedures of popular Flow Lipman et al. (2023): Liu et al. 2023 Albergo Vanden- Eijnden (2023) or Bridge Matching Shi et al. (2023).
Compute C Ex [x( [[
2:
Optimize min. C;
acc Update 8 by using 26
The ground-truth E(x) is estimated with Eq. 77. Specifically, we approximate the field by Monte Carlo samples from the given transport plan π. The field between paired quarks in a batch Ex,,x is determined according to the recipe described in $3.4. Analogously to EFM and PFGM, We learn fo(.) by minimizing the squared error difference between the normalized ground truth E(x) and the predictions [x) over the parameters of the neural network with SGD, i.e., the learning objective is
E
Exll.fo
11 min.
(11)
Inference. After learning the normalized vec-
Algorithm 2 IFM Sampling
tor field E(-)
[E[-]I with a neural network fe(-).
Input: samples X, from P P(x,)8()) step size Ar > 0;
we simulate the movement between hyper- planes to transfer data from IP(X,) to Q(x,). A straightforward approach for this is to run an ODE solver for equation [ However, one needs a right stopping time for the ODE solver since the arrival time may differ for differ- ent field lines. In order to find it, we follow the idea of (Xu al., [2022] Kolesov et al. 2025) and use an equivalent ODE solver with x evolving with the extended variable
The learned field fä (-) : :R D+1 + n+!;
Output: samples &, from ((xx)))(== L)
Set Xo &
for T E (0, 27,2AT.
Calculate fo (x,) L- Ar) do
Xr+ + (xx )2 + + 2 (x Ar]
c XT.
Plao
ky- 7lg
dx = dx dt
1)dz = (E, (X)E (x), 1)dz
dt dz
(12)
E,
E
1)dz ~ fo(8)a 1)dz,
[x(x)=
where we denote (x) as (x), fo (x):) and E(x) equals(E,, (x)). ).)..n)))). In the new ODE (2). we replace the time variable / with the physically meaningful variable 2 setting the explicit start
(a)
(b) I 40
Figure 7: Interaction field line structure for the
Gaussian- SSiiss Roll experiment with L
(z = 0) and the end (z L) conditions. We
6, 40. Our IFM with minibatch OT plan.
start with samples from (x(x)) i.e., when 2 = O. Then, arrive the data distribution Q(x;) when reaches T. during the ODE simulation.
7

---

Published as a conference paper at ICLR 2026
We emphasize, that in EFM such movement does not always realize transport between distributions
properly due to the backward-orienter lines and line termination problem (*2.3). In contrast, in our
implementation of IFM ($3.4), such integration theoretically provably translates P to Q. All the ingredients for training and inference in our method are described in Algorithms and 2 where we summarize the learning and the inference procedures.
EXPERIMENTAL ILLUSTRATIONS
Here we show the proof-of-concer experiments with our IFM method. We show a 2-dimensional illustrative experiment ($..1), image generation ($4.3) and image-to-image translation experiments
(4..)) We give technical details of the implementation of the experiments in Appendix B We
provide a study of the sensitivity of our model to the choice of hyperparameters in Appendix
4.1 GAUSSIAN TO SWISS ROLL
Xg [(Xa)
X4 - (1xa)
y 7(xg)
y- T(x)
(a) Samples from F(x,), ). placed on (b) Samples from placed the hyperplane O. the right hyperplane
placed (c) Mapped samples by Txx,) (d) Mapped samples by T (x,,)
distance
distance 40.
Figure 8: Illustrative 2D Gaussian- -Saluun---iiss Roll: Input and target distributions P(x,) and Q (x,) with the transfer results learned by our IFM (with minibatch OT plan) for distances L 6 and L 40.
An intuitive initial test to validate the method involves transferring between distributions with visually comparable densities. We use 2D zero-mean Gaussian distribution with identity covariance matrix as IP( and a Swiss Roll distribution as Q(x,). Their respective visualizations appear in Figs. 8a and 8b In Figs. 8c and 8d show the points T T(x,) obtained by moving along the lines of our interaction field realization with minibatch OT plan (Tong et al. 2023). Fig, 8c corresponds to the plate distance L = 6, and Fig. corresponds to L = 40. It can be seen that there are no sigmificant differences due the choice of the hyperparameter T., while in EFM the choice of large 7. lead to failure on the same experiment, namely, it failed to accurately map P to Q see (Kolesov et al. 2025 $4.2). Fig. shows the 3D structure of the field lines for L = 6 and I. 40. It can be seen that the lines are almost straight over the entire range of values and depend weakly on L. Note that in EFM the field lines were significantly curved at large values of L (Kolesov et al. 2025 55), see Fig. 7
4.2 IMAGE GENERATION
We consider the generative task on the multimodal 32 32 CIFAR- 10 dataset and the high-dimensional 64 x 64 CelebA faces dataset. In this experiment, we place noise images from Dddimensional multivariate normal distribution N (0, 4 IDDD on the left hyperplane z = O, where D) = 3 x 32 x 32 for CIFAR-IO and D 3 64 64 for CelebA. Images from the CIFAR-10 and CelebA datasets are placed on the right hyperplane interaction field between the hyperplanes using independent transport plans.
20. In accordance with our Algorithm 1, we learn the normalized
For completeness, we compare our approach not only with previous electrostatic-basee approaches such as EFM (Kolesov et al., 2025), PFGM et al. 2022) and PFGM++ (Xu et al. (023), but also with modern flow- FM (Lipman et al.) (023), diffusion-based DDPM (Ho et al.) 2020), and adversarial approaches such as StyleGAN Karras et al. 2020). Our method, IFM, performs competitively with well-known state-of- the-hhh-a art approaches in terms of qualitative results (see Figs. Paland 19)). while EFM fails to generate samples for the 64 x 64 CelebA dataset (see Fig. 9b). We quantitatively evaluate method's 's performance by reporting FID in Table 1.
8

---

Published as a conference paper at ICLR 2026
Dataset/ Method
IFM (our)
EFM
PFGM+
PFGM
FM
DDPM
StyleGAN
CIFAR-10 (32x32)
2.28
2.62
2.15
2.76
2.99
3.12
2,48
Celeba (64x64)
3.07
>100
2.89
3.95
14.45
12.26
3.68
Table 1: Image Generation: FIDI score on 32x32 CIFAR-10 and 64x64 Celeba faces datasets for
our IFM, EFM, PFGM & PFGM++, flow matching (FM), diffusion (DDPM) and StyleGAN.
Additional qualitative results for other image generation tasks (128x128 CelebA dataset and conditional image generation CIFAR-10) are provided in Appendices D Dand F
Computational efficiency. Training of our IFM takes less than 10 hours on single NVIDIA A 100 GPU (30 GB VRAM) for the 32x32 and 64x64 resolution datasets, and less than 30 hours for the 128x128 resolution dataset. Our IFM shares the same architecture the closest competitors: EFM, PFGM/PFGM++, DDPM, and FM. We also use the same Euler- ODE solver and 100 evaluation steps for each method to ensure fair comparison. Therefore, the inference speed and memory usage is identical for all these methods. For completeness, we report the inference speed to generate different batch sizes of images, including a single image (i.e., batch size=1), see Table In particular, the peak GPU memory usage for all methods is approximately 8, 10 and 16 GB during inference with a batch size of 128 for 32x32, 64x64 and 128x 128 datasets, respectively.
Dataset / Batch Size
256
128
64
16
CIFAR-10 (32x32)
10.93
5.74
1.63
0.82
0.7
Celeba (64x64)
36.81
18.45
8.5
2.93
0.97
Celeba (128x122 128)
63.28
32.27
14.87
4.06
1.75
Table 2: The inference time (in seconds) of our IFM with different batch sizes [B] in generation.
4.3 IMAGE-TO-IMAGE TRANSLATION
Following (Zhu et al. 2017 Kolesov al. 2025 also consider an unpaired image translation task
with two scenarios: translating 32 x 32 colored MNIST digits from '2' to '3' (2-3) and translating 64 x 64 scenes from Winter to Summer (W+S). The placement of images on the hyperplanes follows the same setup as the $4.2 We learn the normalized field between the plates using both independent and mini-batch optimal transport plans. Our IFM and IFM-MB approaches effectively preserve shapes (with IFM-MB performing slightly better due to the use of the transport plan) and changes the styles of the initial images (see Figs. [T0a] and TOb). For completeness, we compare IFM and IFM-MB with popular image-to-image translation methods, including Flow Matching, diffusion- (Ho ct al. 2020 DDIB), adversarial (Zhu et al. 2017 CycleGAN), and EFM. We
PM]Duuss
FRM
PFGM
PFGM
FM
DDPM
StyleGAN
IFM:Our)
FFM
PGG
FFGH
FM
DDPM
StyleGAN
(a) CIFAR-1O 32x32. 32x32.
(b) Celeba 64x64.
Figure 9: Image Generation: Samples obtained by IFM(ours) with the independent plan, electrostatic- based approaches EFM and PFGM&PFGM++. blow- FM, diffusion based DDPM and StyleGAN

---

Published as a conference paper at ICLR 2026
evaluate the performance of these methods using the CMMD Yan et al. (2022) reported in Table We also add discussion the distinctions between our approach and FM in the Appendix Fl.
Int
MBIOu (FMICU))
EFM
FM
CycleGAV
DDIB
Inst
FM MB IFMiCuri
EFM
FM
CycleGAN
DDIB
23
3
3
3
3
3
23
3
3
3
3
3
23
3
3
3
3
2333 3
7
3
3
23 3
3
3
3
(a) Colored digits '2' + Colored digits '3'.
(b) Winter Summer.
Figure 10: Image Translation: Samples obtained by (F with/without the minibatch plan.
clectrostatic-base approach EFM, flow-based FM, diffusion-based DDIB and adversarial CycleGAN.
Dataset / Method
IFM-MB (our)
IFM (our)
EFM
FM
CycleGAN
DDIB
22 + '3' (32x32)
0.87
0.95
0.93
1.06
0.90
0.96
W +S (64x64)
1.13
1.25
>1
>1
1.33
1.39
Table 3: Unpaired Image Translation: CMMDJ W-S and colored difits '2' for our IFM, EFM, FM, CycleGAN and DDIB.
-'""
Computational efficiency of our method in translation is almost the same as in generation ($4.2) Also, the core implementation components of IFM (artitectures, number of steps ODE solver, etc.) are the same as in generation and are again identical to the closest ODE-based competitors EFM & FM to ensure the fair comparison. Therefore, the runtime memory usage is the same for all these mentioned approaches. For completeness, we report the inference time of IFM in Table 2 below.
Dataset / Batch Size
256
128
64
16
1
MNIST 32'-'3" (32x32)
10.95
5.73
1.61
0.85
0.73
S (64x64)
36.84
18.44
8.53
4.05
1.76
Table The inference time (in seconds) of our IFM with different batch sizes [B] in translation.
DISCUSSION
Our proposed IFM method is generalization of EFM allowing using rather general interaction fields for distribution transfer. Our implementation of the particular IFM field is just one of many possible realizations. The search for more optimal realization is a difficult task and is a promising subject of future research that opens opportunities for the further development of electrostatic-inspiee models.
Our IFM overcomes major limitations prior EFM method ($2.3): backward-oriene -rrented field lines, line termination problems, and training volume selection issues. Moreover, due to our field's specific structure, we also address the high-dimensionality challenge and the associated numerical instability that affects EFM and PFGM due the Coulomb factor 1/ I
(see Appendix Ch.
Impact statement. Our paper presents work with a goal to advance ML. There are many potential socictal consequences of our work, none of which we feel must be specifically highlighted here.
Reproducibility, We provide the experimental details in Appendix and the code to reproduce the conducted experiments in the supplementary materials (see README.md).
Acknowledgment. The work was supported by the grant for research centers in the field of AI provided by the Ministry of Economic Development of the Russian Federation in accordance with the agreement Sokolov for providing feedback and suggestions for improving the proofs and clarity of our paper.
and the agreement No Nv139-10-220 10-2025--33 We thank Kirill
10

---

Published as a conference paper at ICLR 2026
LLM Usage. Large Language Models (LLMs) were used only assist with rephrasing sentences and improving the clarity of the text. All scientific content, results, and interpretations this paper were developed solely by the authors.
REFERENCES
Michael Samuel Albergo and Eric Vanden-Fijnden. Building normalizing flows with stochastic interpolants. In The Eleventh International Conference on Learning Representations, 2023.
Xiao Cao and Shenghui Zhao. Pfgm++ combined with stochastic regeneration for speech enhance- ment. In 2024 9th International Conference on Signal and Image Processing (ICSIP), pp. 267-271.
IEEE, 2024.
Xiao Cao, Shenghui Zhao, Yajing Hu, and Jing Wang. Sr-pfgm+ based consistency model for speech enhancement. In 2024 IEEE International Conference on Signal. Information and Data Processing (ICSIDP), pp. 1-5. IEEE, 2024.
Francisco Caruso, Vitor Oguri, and Felipe Silveira. Still learning about space dimensionality: From the description of hydrogen atom by generalized wave equation for dimensions d 2 3. American Journal of Physics, 91(2):153 158, 2023.
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-6851, 2020.
Tero Karras, Samuli Laine, Miika Aittala, Janne Hellsten, Jaakko Lehtinen, and Timo Aila. Analyzing and improving the image quality of stylegan. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 1110-8119, 2020.
Diederik Kingma and Jimmy Lei Ba. Adam: method for stochastic gradient descent. ICLR: international conference on learning representations, pp. 1-15. ICLR US., 2015.
Alexander Kolesov, Stepan Manukhov, Vladimir V Palyulin, and Alexander Korotin. Field matching: an electrostatic paradigm to generate and transfer data. In Forty- International Conference on Machine Learning, 2025. UtLpttps::// httpt::::///peeeevee
net. furum? d= 1xyiiilyvv
L. D. Landau and M. Lifshitz. The Classical Theory of Fields (Volume of A Course of Theoretical Physics). Pergamon Press, 1971.
Yaron Lipman, Ricky TQ Chen, Heli Ben- Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. The Eleventh International Conference on Learning Representations, 2023.
Xingchao Liu, Chengyue Gong, and qiang liu. Flow straight and fast: Learning generate and transfer data with rectified flow. In The Eleventh International Conference on Learning Representations. 2023.
Aram-Alexandrr Pooladian, Heli Ben-Hamu, Carles Domingo-Enricn -nniicg Brandon Amos, Yaron Lipman, and Ricky TQ Chen. Multisample flow matching: Straightening flows with minibatch couplings. In International Conference on Machine Learning, pp. 28100-28127, PMLR, 2023.
Fernando Quevedo and Andreas Schachner. Cambridge Lectures on The Standard Model. CERN- TTH-2024-006 1064-00,7770 2024.
Yuyang Shi, Valentin De Bortoli, Andrew Campbell, and Arnaud Doucet. Diffusion schrödinger bridge matching. Advances in Neural Information Processing Systems, 36:62183-62232 2023.
Jascha Sohl-Dickster Dick-issttein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. International conference on machine learning, pp. 2256-2265. PMLR, 2015.
Alexander Tong, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector- Brotor, Kilian Fatras, Guy Wolf, and Yoshua Bengio. Conditional flow matching: Simulation free dynamic optimal transport. arXiv preprint 202iv22302.00482. 2(3), 2023.

---

Published as a conference paper at ICLR 2026
Yilun Xu, Ziming Liu, Max Tegmark, and Tommi Jaakkola. Poisson flow generative models. In Proceedings of the 36th International Conference on Neural Information Processing Systems, pp. 16782-16795, 2022.
Yilun Xu, Ziming Liu, Yonglong Tian, Shangyuan Tong, Max Tegmark, and Tommi Jaakkola. Pfgm++: Unlocking the potential of physics-inspired generative models. In International Confer- ence on Machine Learning, pp. 18566-38599) PMLR, 2023.
Shifu Yan, Caihua Shan, Wenyi Yang. Bixiong Xu, Dongsheng Li, Lili Qiu, Jie Tong, and Qi Zhang.
Cmmd: Cross-metric multi-dimensional root cause analysis. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 4310-4320, 2022.
Jun- Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-imag translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pp. 2223-2232, 2017.
12

---

Published as a conference paper at ICLR 2026
A INTERACTION FIELD MATCHING: PROPERTIES AND PROOFS
A.1 PROPERTIES OF INTERACTION FIELD LINES
In this section we formulate several properties the interaction field which are essential for the proof of the main theorem. First of all, let us formulate the notion of field flux which has been introduced intuitively in the main text.
Definition A.1. (Field flux). Consider an element of area dS. The field flux E through this element is called d&p = E dS. If we need to calculate the field flux through finite surface Σ, then the field flux is
dΦ
E...
(13)
Intuitively, field flux indicates how many field lines pass through surface Σ. The greater the flux, the higher the number of lines passing through a given area.
Remark, For the closed surfaces, we assume that the normal is always directed outward.
Lemma A.1 (Generalized Gauss theorem). Let and Q be two probability distributions on RDD, and let sPpp(P) and supp( QQ) denote their respective compact supports. These supports are lo- cated on the planes 2 O and z = L in the extended space IR DD-1 and the distributions satisfy (mpp(() Q(xq)dxg. Let (xg Ex,xx
S Supppry PP dxx, by a pair of unit quark and unit antiquark satisfying properties 1-3 of $3.2with with the transport plan N (Xqq Xq). Let M C RD+1 be open bounded set and let 0M be its boundary (a piecewise smooth hypersurface) such that M n supp((P) + Ø and M n suupp( Q = Ø. Then
x, (x) be an interaction field produced
JJ 0M
E ds = to PXX))xx
(14)
where Do = Noo Egg
dS is the field flux from a single unit qq-pain and (xx) (x)))(2).
Proof. Substituting the explicit expression for the interaction field E(X) we obtain:
[]
ds = -f
(/ E Egg (Xg: X4) dxx
ds
8M
8M
(X X dx dxp
Eng ds
5
GM
| s
(xx: Xxy dx
F [x, E supp(P) MJ
(15)
dxy I (upp(()) (x.
20
Φ Px())dxx
where S supp(P)) x supp(Q) denotes the Cartesian product of the supports, and θ(x, E supp(P)) n M) is the indicator function equal to 1 if X.y E supp(IP) n M, and 0 otherwise.
The chain of equalities proceeds as follows: first, we substitute the definition of E(x) from equation 7 then we interchange the order of integration (justified by the continuity of and Eqq and the compactness of their supports); next, we use that the flux of single pair Eqq through 0M equals Po
*We assume ($2.2) that the flux E,ÿ is proportional to the charge of the quark that creates field E,g and does not depend on the relative position of the antuquanti pair. That is, all other things being equal, the replacement e.g. q + 2g will lead to &gg - 24 Therefore, for any pair qů with the same unit charge (which create elementary field Egg), the flux 4gg t.1 4. will be the same for all pairs
13

---

Here 1 since the height of the cylinder under consideration be made as small as want (in- finitesimal higher order than dS). d&- = -E dS has negative sign due the fact that the normal to the Published as a conference paper at ICLR 2026
precisely when the quark lies inside M, which is expressed by the indicator; finally, integrate Xg to obtain the required result Po SM P(x)dx
Remark. If M contains a part of the distribution Q(-) but does not contain P(.). then the statement of the theorem will be written as follows:
E .edds ds - Po Q(x)dx.
(16)
AM
Corollary A.2. For any point x in the support of distribution P(:):
(x) 15 (x) 20 Pxx)..
(17)
where E (x) (x,
Proof. Consider volume dS E supp P(.). Consider a closed surface, cylinder with infinitesimal indentation in different directions the plane = = 0, see Fig. 11
The flux through this surface consists of three summands: the d flux in the positive direction of the z-axis, the d$ flux the negative direction of the z-axis, and the d Im flux through the lateral surface:
d d + + dl + d = E E, d8 + 0.
(18)
Pds
closed surface is directed outward, i.e. in the opposite direction from the axis
E
Then, due to the generalized Gauss's theorem:
dΦ
Eg dS
d$
E dS
d (E Eg dd 4o PdS (19)
lat
which proves the corollary.
Lemma A.3 (On field lines). Let P().) and Q(-) be two (discrete or continuous) distributions corresponding to quarks and antiquarks. Let these distributions have
Figure 11: Considered area.
compact support and satisfy I P(xx)dx, { Q(xx) Let the field of the unit quark-antiquark pair Ex,xx (x) Egg(( (x) start Xg and end Xq (Property 1), and conserve flux along the current tube (Property 2). Then the total field 7 from all quarks and antiquarks satisfies:
(a) Its lines start P() and end at Q(-), except perhaps for the number of lines of zero flux (b) It conserves flux along the current tubes.
Proof. Let begin with the proof of the second property. Let the field E ox and Eng dS const. Then:
/ E ds =
/ Toddxxdxa
Egg dS
Togdxxdxx const const. 1 const. (20)
Let us now prove the first property. Suppose the opposite. Let there be lines starting at IP but not ending at (the case of lines not starting IP but ending at Q can be considered similarly).
14

---

Published as a conference paper at ICLR 2026
Consider qq-pair. The field Egg of this pair by assumption of the lemma starts and ends on continuous, and therefore cannot go to infinity. Therefore, the flux through any E-area infinitely distant from qq-pair must be zero
$ 99
Eug
ds + 0, 0,/|/ + 30.
(21)
Then, if I--) and Q(-) have compact support, the field flux from all quarks will also be zero:
& ( E.S. J
EmnTTodxxxxxx dS =
-f Ton D dx + 0, 0,11x|1 - oc. (22)
Therefore, if the lines under consideration go to infinity, their flux is zero
Let now consider lines ending not at infinity and not the target distribution (i.e., at some intermediate points |
At the stopping point E = 0 (otherwise it is not stopping point). But then the
total flux produced by such lines:
/
E.Edd ds
(23)
which proves the statement.
A.2 DEFINITION OF STOCHASTIC MAP
Movement from one distribution to the other is carried out along the field strength lines. In this section, we rigorously define this movement using a stochastic map T, taking into account existence of two series of lines forward-oriented and backward-oriented
We define the stochastic forward map TF from supp(P) to (Q) through forward-oriented field lines. For this, we consider a point X,- = E),E + 0+ slightly shifted in the direction of the second plate. Let us move along the corresponding field line by integrating dx((t) = E(t
Sooner later, such movement leads to the intersection of the plane 0 or L. At this moment, the question arises of whether to continue or stop the movement. If the movement continues, it will L, is intersected again. Then, decision must again be made to
proceed until the plane 0 stop or to go further. This procedure must be continued until we reach the final stopping point. Let us denote the intersection points follows:
X,
(N)
(24)
At each of these points, is necessary stop with probability v and continue movement with probability - xxx()) ), where
1
if E (belore)
and E (llee)
have opposite signs
(x(())
if E after)
(eforc)
(x
| (
if (aller
before)
E botom)
(25)
"The difference & with the 4, here is that 2 is an open surface infinitely distant from the qğ -air, while Po the flux through closed surface 0M containing charge g and not containing charge q. In other words, Po denotes the total field flux between through some infinitely distant surface.
antuquan pair of unit charge, while & defines the field flux
Note that there only two possible options either the field line goes to infinity, does not go to infinity, that is, stops somewhere intermediate point.
15

---

Published as a conference paper at ICLR 2026
where E ( (x and E.
the values of the z-component of the field immediately before and immediately after intersecting the plane at point F respectively.
(after)
To understand the meaning of this probability, note that if E (befor)) (x and E (ateer)) (x(2 have opposite signs, then further movement along the field lines is impossible (and thus we have arrived at
the final point x on the target distribution). If they have the same sign, then further movement along the field lines is possible.
If further movement is possible, two situations arise: either F5g (after) (())
|F (hefore)
X or
L (belloe)) (x()). In the first case, after crossing the plane, the field magnitude (and consequently the flux) increases therefore the stopping probability is set zero. the flux magnitude becomes smaller after crossing, it means that one must stop with some probability and continue with another. Lemma X explains why this particular probability value chosen for the latter situation.
| (after)) (1) F
The stochastic backward map Ta is constructed similarly using left limit & 0 and backward- oriented field lines.
The complete transport T is then described by the random variable:
(Tx(x,) with probability ((xx);;
T(x,) capturing both forward and backward trajectory endpoints for each x,, E supp(IP) with probabilities ((x,) and 1 (xx,), where
(26)
(X,, with probability 1 - -)((x),
0,
Eg <0
plx,))
1,
E >0 0
(27)
E
otherwise.
0+ are the left and right limits of the field value at the point in direction. The meaning of this probability is follows. Value μ(x,) allows one to choose forward or backward sets lines with a probability proportional to the field flux in the corresponding direction (i.e., proportional to E and [E respectively). At the same time, if it is impossible to move forward (E++ < 0), the map T3 is chosen (j((x)) = 0), and if is impossible to move backward (E, 0), the map TF is chosen (x(X,) 1).
Here E
A.3 IFM MAIN THEOREM PROOF
Lemma A.4 (First lemma on the flow). Let TP(.), Q(-) be two D-dimensional data distributions having compact support, located on the planes z O and
in D+1 respectively. Let (K.g H be sample of points distributed over P. Let dS be an element of D-dimensional area on the distribution of P (dS E supp PP. Let the field near the element dS have different signs: E and E_ < 0. Let dn be the number of points from the sample that fall in the volume dS. Let dn dny + du where day is the number of points from dS that correspond to the mapping Tr (ie., movement along forward-oriented lines), and du corresponds to TB- Then:
dnF
EtdS
d F
12+00
P
P
(28)
dn
dS
d n
a.s.
n
n-000
where (... n.s.
denotes the almost sure convergence.
Proof. According to the multiplication rule of probability and the law of large numbers:
dnp a.s.
(probability of choosing TF): (probability of falling dS)
n
(29)
E
E + |E 1)S
E:ds
d F
= (x() P(XdSS=
E
E=
20
20
40
16

---

Published as a conference paper at ICLR 2026
In the second equality, the definitions of probability u(-), see Eq. (7 and Corollary A.2 The case dn B is proved similarly.
used.
Lemma A.5 (Second lemma on the flow). Let IP, Q, (F 2 -i= the Lemma A.4 Let EJ and E have the same sign near dS (i.e., either simultaneously E > 0 or
dS, dn have the same meaning in
simultaneously E 0). Then
dn
d after
d before
0.5
(30)
12-100
Po
Do
where d before the field flux through the current tube supported on dS immediately before crossing the plane dS E supp P(.), and d after is the flux after crossing.
Remark. This statement implies that when the field crosses the plane containing a charge (proportional to dn/n), the field flux must increase by 20 dn/n.
Proof. For clarity, consider the case E > O, E > when fa( u(x,) = and motion processes only along the forward-oriented lines, corresponding to the mapping TF.
By the probability product theorem, the strong law of large numbers, Corollary A.2, and the definition of flux:
dn + (x))((x) = 1 P(x)dS
E E 4
d
daPater
d'Pre
dS
(31)
4
40
Lemmas A and A.5 address the behavior near the distribution P. Similar statements are valid for the behavior near Q. When moving along field lines, we eventually reach the plane = L. At this point two different scenarios may occur:
1. (L) and Eg (L) have opposite signs. Then the field line motion terminates in this case. 2. (L) and Eg (L) have the same sign. Then portion da' of lines must terminate, while others continue.
This portion dn' can be found from the line termination property in Q. Lemma A.6 (Line Termination). D (L) and E (L) have the same sign upon crossing L. the number of lines terminating on = L satisfies:
dn'
ddh Dafier
d before
(32)
P
Remark. When the field crosses the plane Q containing a charge (proportional to dn' /n), the field flux must decrease by Py dn'/ /nn
Proof. Consider the current tube before it intersects the plane = = L. Let us denote the number of L. some of the lines dn' stop moving, while
lines inside dribefore- As result of the intersection some of the lines dnaeer continue moving. In view of the first Lemma A.4 on flow, as well as the conservation of flow inside the current tube (Property 2 in 632
d'bbfooee
a.s.
d beffoee
(33)
n
Then, by virtue of the law of large numbers and the fact that drabefore dn' + draatiers we have:
dn'
dribefore u.s.
(probability of termination)
v(x)
d
100
n n-
(34)
E
E E. dS' =(E )dS' = -d + de
E
17

---

Published as a conference paper at ICLR 2026
We now proceed to prove the main theorem.
Theorem A.7 (Interaction Field Matching). Let P(x,) and (xxa) be two data distributions that have compact support. Let Xg be distributed over P(xg). Then xg = T(x.) is distributed over (xx) almost surely:
If x4 P(x,) + Txx)) = x Q(x.).
(35)
Proof. Let {xq. 1 be points distributed according to IP. Moving along the field lines via mapping T, we obtain points Xg. Tx,,) in distribution Q
Consider a D-dimensional area element dS' C supp Q. Let dn' be the number of points X in this area. Define the sample:
dn
Qn.5s'
(36)
The aim is to prove
Q 2--0 Q
(37)
2=L
The points dn' arrive via forward or backward directions:
IdS'
dn' dn'p + dn'g.
(38)
dn'p
dn B
dn
Consider dn'r and its associated flux tube. Traverse this tube inversely along the field lines until stopping at P. During this motion, multiple crossings of and/or 2 may occur. Denote the intersection points:
(xq)
Figure 12: Points corre- (39) sponding to forward and backward lines.
X
XN-1 + XN Xq.
Their corresponding area elements are
ds' = dSg + dSi + ..... dSN + dSN = dS.
(40)
Point counts in these areas read
dn' = dno + dny + .... dn dn dn,
(41)
where dn& (k = O..., N + 1) is number of points from sample fxg. 1 or from map (T ) inside the volume dS, near point x₃, that corresponds considered motion inside current tube.
The drgg are not arbitrary but related by flux conservation. Only the charged planes (= 0 or L.)
can alter the count:
At 2; = 0: Line count increases by dn;
At 2 = L: Line count decreases by dn;
Mathematically:
N
fn dn, = O,
(42)
1=0
where
0 if 0,
f
(43)
if 2; L.
Due to the first Lemma flow A.4
18

---

Published as a conference paper at ICLR 2026
dny
dn
d N
a.s.
(44)
71
n-
40
40
Due to the second Lemma A.5
on the flow and because of the line termination Lemma
A.3
(-1) dn,
d Patters
d oeforre
(45)
n 70
P0
Po
According to the law of conservation of flux along the tube (Lemma
A.3
d anterr ddPbefore,
(46)
Whence we obtain a chain of equalities:
N+1
dn'p
+ (-1) dn1
dnN
+ dn
0 1h dng
(-1)
n
k=0
dn'p
dni + + (-1)JN (-1)
dnN dn as.
(47)
n
n
2
- dD after.1 ddPrctore.1 + + dlp aaean dl defoorre + d4 N+1++1 = as.
d'aanerr1 + 0 + 0 = 0dd''p.
Consequently,
dn'y
d//r
as
(48)
71
n-500
40
Similarly, it can be proven that
dn'p d
(49)
n
11-000
40
Then, by virtue of the generalized Gauss's 's theorem (Lemma
A.1
we finally have
dn'
dn'p
dn'g
d
d
d''
+
a.s.
+
QdS.
(50)
71
n-0000
40
d
This completes the proof.
A.4 INTERACTION FIELD REALIZATION
Here we formulate an algorithm for computing our constructed field which inpired by strong interaction in physics at an arbitrary point X E D+1 with the quark and the antiquark being at X, and X2 (see Fig. 6).
Symmetric case. Let quark be located at the origin: x, (0, 0) DR1,, and the antiquark q at the point X2 (0, L) R D+1 The arbitrary point space can be written
(x1, 2)
x + ze,, where _ D+1 is the component the vector x orthogonal the z-axis. We introduce the following string hyperparameters (Fig. 6a):
is the effective width of the string in the cross section.
d is the size of the region of the string in which the field lines will curve toward the quark (antiquark). Thus, in the interval E [d, L d the field lines are straight, and in the regions 2 E [0, dl and z E [L d, L] the lines will be curved. Value म 7/2d also introduced.
We define the dependence of the effective string width (2) the coordinate follows:
19

---

Published as a conference paper at ICLR 2026
Algorithm 3 Interaction field calculation
Input: Positions of quark and antiquark: X,, X,y E IR D+1 with 2n = 2g
Arbitrary point X E RDD1
String hyperparameters: oo d /
Output: The interaction field Egg
Algorithm:
Calculate the vector connecting the quarks: F- X- X2 E R D+1
Calculate the unit direction vector corresponding to it: e = [Ti E D+1
Calculate the vector of shift of the point x from the axis of the string:
p = x x + (Rg R E E
where is corresponding coordinate of point x
Calculate _ = || [PTT
Calculate the string width (=) according to 51
Calculate the angle ax 1. 2) according (53)
Calculate the value of field (( according to (54)
Calculate the direction nxx) cos ((x 2) e +imaa(z1)2) te E R-1
Return: Egg(X) (X) = E z)n((1,
oo (kz),
E [0, d].
2 E [d, L d]
00.
(51)
oo sin (k(1 =)),
2 E [.. - d, d,
0,
otherwise.
The field direction nx) at the point X is defined as:
x cos a
+ sin to E ER. D+1
(52)
where e., Cn are the unit vectors along the 2-axis and along the vector X respectively, i.e., ej x ||. a a(x1,2) is the angle between the field direction at a given point and the 2-axis. This angle determined from the following considerations. Let (²(=') be the field line parallel to the level a=)) (i.e. Az' > (=()//(((=)) = const) which passes through the point (x1, 2), i.e.,
(===))==== = x (x1,2) 2). Then a is determined by tan de
da'
aarcann (rctan((=__co((())))) 0,
E [0, d),
a = a(x1,2)
E [d, d)
(53)
cot(k(L 2))),
E LL d, L).
We define the field strength value as the product of the Gaussian distribution in the radial direction and normalization factor that keeps the interaction field flux invariant along the tube:
2(I1,2)= = cxp 20(
(54)
(22) cos a(=1,2)
Shifted case. In the case where the quarks are in the shifted positions X, and Kg. we use field shift parallel to the planes 0 and I,, as shown in Fig. 6bb We use the shift and not the rotation of the string with the aim not generating backward-oriented lines and lines traversing the region L. The detailed algorithm for calculating the field is formulated in Algorithm
A.5 PROOF OF PROPERTIES OF INTERACTION FIELD REALIZATION
Theorem A.8 (Properties of our interaction field realization). Our realization of the interaction field E(x) satisfies the fundamental Properties 1-2 in 43.2 with following additional characteristics:
20

---

flux tube Published as a conference paper at ICLR 2026
Z-Axis caging: Field lines never extend beyond L.
Unidirectional Flow: No backward-oriented field lines exist.
Centrosymmetrical arrangement: E(x) 2))
Radial Decay: Monotonic decrease in field strength away from axis:
@|
I 0
with
lim
E(x)
+000
Axial Aligument: Field becomes parallel to the string axis in middle region:
E(r1,2) || dz for 2 e d, L d...
Proof. The interaction field starts at quark and ends at antiquark (Property 153.2) due to the fact that the field direction n(x) cos (x o'g + sin ((21,, - to is tangent the curve x'1 (z"), which by construction begins at X, and ends X.y
Consider an infinitesimal current tube connecting a quark and an antiquark. The surface bounding this tube is parallel to the field line x'1 (="). Along the field line by construction /a(=)) K const. In (D+ 10++1--ah dimmenssiol space, the area element dS orthogonal to the z-axis is dS 2 dr. Therefore, due to the definition the flux and the explicit expression for E( 2) 54 we have
22
dΦ = E dS
EdS CO8 exp
20)
If dx
-0S a
ol2) cos
(55)
D-1 de const.
exp
2
Therefore, Property 33. satisfied.
The Z-Axis caging property is satisfied because a(= > L) 0. The Unidirectional Flow property satisfied due to a < L) = O. The Cylindrical Symmetry property is satisfied because E(x) 2) Radial decay property is satisfied because of the explicit formula (5)) for E
Finally, the Axial alignment property is satisfied because a<22 E [d, d)
Remark. A crucial element in the flux conservation proof is the factor 1/) in the definition of the IFM field (see (54)). The intuition behind this factor can be explained as follows: any flux tube must narrow a point as it approaches charge. Consequently, the cross-section (which proportional to (2))) must also decrease. Flux conservation can only be maintained by proportional increase in the field strength, see Figure
below.
S
R D+1
S
S
E S E S1 const
Xg
ST D
Figure 13: An illustration to the flux conservation property. To maintain the field flux conservation within a tube as narrow,, a proportional increase in field strength required.
In simpler terms, the closer is to charge, the stronger the field must be.
21

---

Published as a conference paper at ICLR 2026
B EXPERIMENTAL DETAILS
We aggregate the hyper-parameters of our Algorithm I for different experiments in the Table 5 We base our code for the experiments on EFM's code
com http:/hhwb...oo/stttkotooooeeeee https::/gginne www.ahhww..commssstkeeeeeeee
F FieldMatching Our code is available at
InteractionE htw.:oonneeeMMMMataiihh..
Experiment
D
Batch Size
L
LR
plan
Gaussian Swiss-roll
2
1024
[6. 40)
O.1.
2e-4
[Tnd, MR]
CIFAR- 10
0.5L
ColebA 64X64 Generation 4.2
12288
128
20
0.11
0.25, 0.1.3.
2c-4
Ind
MNIST digis 2-+3 Translation 84.3
3U72
128
20
0.52
2e-4
[nd. MB
Winter- Translation
12288
128
0.5L
20-4
[[nd, MB]
CelebA 128x Generation App.
49152
128
20
0.5T.
2r.4
Ind
Conditional CIFAR 10 generation App.
E
3072
128
20
0.5L
2e-4
Ind
Table 5: Hyper-paramettrs of Alg. for the experiments, where D is the dimensionality of task, is the distance betwenn plates, is the effective width, d is the characteristic distance (see Fig 6a).
In the case of the Image experiments (see $.3 and $4.2 we follow (Kolesov et al., 2025 LFM), Xu
ct al. 2022 2023 PFGM/PFGM++) (Ho et al. 2020 DDPM) and (Lipman et al. 2023 FM) and
Exponential Moving Averaging (EMA) technique with the ema rate decay equals 0.99 provide
smooth solution. Also, we use linear scheduler, that grows from to 2e 4 during the first 5000
iterations and decreases monotonically. As for the optimizer, we use Adam optimizer Kingma & Ba (2015) with the learning rate 2e 4 and weight decay equals 1 4.
We compare method with PFGM/PFGM++XXeet al. (2022 2023). whose the source code taken from http://twss....tttc..coo////eees wwt.c///eeeeeeee/ooosooooooooooo
gmpp for running PFGM++ and https for PFGM in our experiments. We follow the proposed values of hyper parameters are uppropriate for us: 5 0.3, le 3. The source code for DDPM is taken from httpt::pp//ot httpt:ttttttthhhl com a- wwwgsssoggsoooeeee sde pytorch
//github.
with paremeerrsetsrss min 0.01, maar = 50, Bmin = 0.1 and 3maz = 20. The source
code for FM is taken
/ ctt faem///ccboookkoo waw.oookreeerrccccceceoo flow wateerrr//llllmtcchhhhaea followmatcchins
with linear interpolant The source code for StyleGAN is taken from
com/
NVLabs/stylegan: ada pytt-pyytoch
C ABLATION STUDY
Our IFM realization is defined by the following hyperparameters: the distance L between plates, the string width ou. and the distance d over which field lines curve toward the charges. In this Appendix, we address the practical selection of these hyperparameters and present an ablation study on how they affect our model's performance. We choose parameters based the following ideas:
Since we learn the normalized field (see 11)), the factor 1/) cancels out. Indeed, let & E R D+1 be a point where we estimate the normalized vector field using B sampled pairs of
quarks and anti-quarks. In accordance with the superposition principle (see (7). the resulting field
is obtained as the average of B independent fields ((x 2) (see (54)) from each pair:
E
B exp(
(0(
n(x())
B
exp(
n =)|
B[I
a(z)o
(=
Therefore, term (2) cancels out completely. The practical choice of the hyperparameter do determined solely by numerical considerations and is usually a 1.
2. The distance d should not be chosen too thort-this complicates data translation via the ODE due to the high curvature of the field lines in the region 2 E d U d, L] In practice, we usually E [0.1L, O.5L..
3. Finally, the distance L does not significantly impact translation quality in our method (see Fig.
7). This is different from EFM, where making too large significantly worsens the results (see
2.3 Our IFM realization is specifically designed to reduce this dependency through straight field
22

---

Published as a conference paper at ICLR 2026
segments for E [d. L d]. where ODE integration follows straight lines. In practice, analogously to EFM, we set L to be the order of the data standard deviation: L /Dp or VDa.
Figure 14
presents a series experiments with different values of the parameter d. can be seen that the generation quality does not significantly depend on this parameter.
(a) L 20, ou 1,d 0.1.L
(b) 20, 1,d
(c) 20, ou 1,d 0.4L
(d) L
20, ou
1,
0.
0.5L
Figure 14: Image Generation on CelebA 64x64: Investigation of generation quality dependence on the model hyperparameter our IFM method.
D ADDITIONAL CELEBA GENERATION EXPERIMENT (128x128)
We also provide more challenging image generation task the 128x128 CelebA dataset. We follow the experimental design from the $4.2 placing the CelebA images and the noise from standard multivariate distribution (0, 1228 128) the left hyperplane (z 0) and the right hyperplane (z = 20), respectively. We present the qualitative results of our IFM in Fig. 15 demonstrating its scalability in high-dimensional spaces.
Figure 15: Image Generation: Samples obtained by IFM (ours) method with the independent transport plan on CelebA dataset with resolution 128x128.
E CONDITIONAL IMAGE GENERATION ON CIFAR-10
Our IFM can be easily adapted to conditional generation tasks. For generating images of a specific class learn conditional vector field E()c).. Specifically, for data sample X, (x,, 0) from e-th class, we sample noised sample x via (10) and approximate E(X]c) by a neural network so (x, c) with the following optimization function over parameters A:
E c)
E. Exje [o (R, c)
12
min.
|[
c)
2
We consider conditional generating task on the 32x32 CIFAR- 10 dataset and demonstrate generated images over each class in Fig. 16
23

---

Published as a conference paper at ICLR 2026
Figure 16: Conditional Image Generation: Samples obtained by conditional vector field B(x]e) of IFM(ours) method on CIFAR- 10 dataset for each class c.
COMPARISON WITH FLOW MATCHING
Our IFM framework offers important advantage compared to Flow Matching (Lipman et al. 2023 Liu et al. 2023 Tong et al. 2023): it enables multi-sample estimation of the field.
In particular, our IFM method approximates the normalized vector field with neural network fe (x), trained with the loss
E
LIFM Ex
(x)
[
which requires an estimate of the ground-trutt vector field Exx). The distribution over points x at which the field learned serves as a hyperparameter. Since the field E(x) is represented using the superposition principle (7), we can estimate it by averaging over fields induced by B batch samples (quark and anti-quark pairs) X, = (xy. 0) and Xg = (x,,, L):
B
E
E 1=1
xx, (x),
B
where each Ex,,xx, (x) admits closed form (see Appendix A.4 Thus, any available number of sample pairs (Xy. Xq) up the entire dataset estimate the ground-truth field and reduce the variance of this Monte Carlo estimator.
In contrast, the Flow Matching (FM) loss is
LPM [, (xo.x))<> [|vo t) (x1 xo)
where X, = [x, (1 t x)x The optimal vector field is v*(,, 1) = E[x: Xg
xx). but this conditional expectation is intractable estimate via Monte Carlo because one cannot easily sample X1, XD conditioned on x,. Thus, during training, one regresses (x t) to its single-sample estimate
vo(x4,t)== 1) x1 Xp-
Therefore, FM estimates the ground-trutt field at each point x, using only one pair (Xo; X1) m. with direct way to reduce the variance of this Monte Carlo estimate.
sum up differences between FM and our IFM in Table
24

---

Published as a conference paper at ICLR 2026
IFM (owrs)
FM
Estimation of held
Mulp-sample: ((X) over pairs (No, (x) ~ x
One-sample: v(x.) over one pair (Na.X1) ~ ST
Training volume
Any: 12 [0. L.]
Restricted: N, (x] (1 <)Ng
Table 6: The differences between our IFM and Flow Matching (FM).
25


Reviews:
[{"review_id": "5hEcPLuS4n", "reviewer": "Reviewer_KeQQ", "summary": "The paper proposes Interaction Field Matching (IFM), a novel physics-inspired framework for data generation and transfer designed to overcome limitations of existing Electrostatic Field Matching (EFM) models. The authors identify issues with EFM such as curved field lines, extension beyond target regions, and insufficient coverage of the target distribution. IFM addresses these problems by introducing a new field structure that results in nearly straight field lines between planes, prevents extension beyond z>L, and adequately covers the entire target distribution. The paper provides theoretical foundations for the method, including proofs of flow conservation properties, and validates it through experiments on multiple datasets. Results show that IFM performs well on datasets like 64×64 CelebA where EFM fails. Additionally, IFM demonstrates competitive performance in generation quality compared to state-of-the-art methods such as PFGM, Flow Matching, DDPM, and StyleGAN.", "questions": "1. How does the small value problem from 1/σ(z)^D practically affect algorithm performance in high dimensions (e.g., D>100)? Have you tried any numerical stability techniques (such as log-space calculations) to mitigate this issue?\n2. Figure 4 shows the impact of different L values on results but does not explain how to select the optimal L value. In practical applications, is there an automatic method to determine L? Is there any relationship between L and dataset characteristics?\n3. How does IFM's training and sampling efficiency compare with other generative models? Could you provide quantitative comparisons with methods like PFGM and DDPM in terms of computational time and memory usage?\n4. How is the \"crucial flow conservation property\" mentioned in Appendix A.4 ensured? Could you explain in more detail why IFM's field structure maintains this property while EFM cannot?\n5. Can the IFM framework be extended to conditional generation tasks? For example, could class information or text descriptions be incorporated into the field structure for conditional image generation?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The paper proposes a novel physics-inspired framework that addresses key limitations of existing EFM methods. IFM produces nearly straight field lines and effectively covers the target distribution, which is significant both theoretically and practically.\n2. The authors provide solid theoretical foundations, including proofs of flow conservation properties and mathematical analysis of field line behavior. Theorems and lemmas in the appendix guarantee the correctness of the method.\n3. The experimental design is comprehensive, comparing not only with EFM, PFGM, and PFGM++ but also with modern flow-based methods (Flow Matching), diffusion-based methods (DDPM), and adversarial approaches (StyleGAN), demonstrating IFM's effectiveness.\n4. Figure 3 clearly illustrates IFM's advantages over EFM, visually explaining why IFM better handles data transfer problems. This visualization is extremely helpful for understanding the method's improvements.\n5. The authors honestly discuss the limitations of their method, particularly the numerical precision issues that may arise in high dimensions (1/σ(z)^D potentially producing values close to machine precision). This transparency enhances the paper's credibility.", "weaknesses": "1. Although the paper mentions potential numerical precision issues in high dimensions, it lacks systematic analysis of their practical impact. In real applications, when dimension D is large, this issue could cause algorithm instability or failure, but the paper does not provide solutions or mitigation strategies.\n2. The experimental section only presents qualitative results (such as Figures 9a and 9b) without quantitative evaluation metrics. In the field of generative models, standard metrics like FID and IS are crucial for objective performance comparison, but the paper does not report these.\n3. The paper does not provide detailed analysis of IFM's computational complexity and runtime. Compared to existing methods, does IFM introduce significant computational overhead? This is crucial for practical applications, but the paper does not provide relevant information.\n4. Insufficient parameter sensitivity analysis. IFM likely depends on key parameters (such as distance L, function σ(z), etc.), but the paper does not systematically study how these parameters affect performance. Figure 4 shows results for different L values, but lacks analysis of other parameters.\n5. The comparison with existing methods is not comprehensive. While the paper mentions comparisons with PFGM and PFGM++, it does not detail the specific theoretical and implementation differences between IFM and these methods, or why IFM performs better in certain cases.\n6. The paper does not explore IFM's applicability to other data types (such as text, audio, or graph-structured data). While the focus is on image generation, discussion of extension to other data modalities would enhance the method's generality.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4p6O6jHdkK", "reviewer": "Reviewer_TnaZ", "summary": "The paper proposes Interaction Field Matching (IFM), a physics-inspired method for data distribution transfer problems. In comparison to Electrostatic Field Matching (EFM), IFM generalized to arbitrary interactions between charges and doesn’t suffer the same problems with tracing backward-oriented field lines. The paper presents the main theorem that movement along interaction field lines provably transfers one data distribution to another. Experiments demonstrate that IFM performs comparably in image generation tasks to well-known approaches such as flow-based models and DDPM.", "questions": "How is this related to kernel-based MMD methods (see above)? It might strengthen the paper to explicitly discuss the connection to MMD and clarify what additional value this perspective of field interactions brings.\n\nAre there guidelines besides the general requirements for choosing the interaction field? Did the authors do ablations on different choices of interaction field?\n\nCan the authors motivate why one would want to use IFM over existing generative frameworks?\n\n[1] Gretton, A., Borgwardt, K. M., Rasch, M. J., Schölkopf, B., & Smola, A. J. A Kernel Two-Sample Test. Journal of Machine Learning Research (JMLR), 2012.", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is well written and easy to follow, with careful explanations of the limitations of EFM (e.g. the line termination problem in Figure 2). It creatively draws from ideas in physics to propose reasonable properties of interaction fields (e.g. the start/termination of lines and flux conservation). It is especially useful that IFM provably transfers one data distribution to the other. From the visualizations in Figure 9 and the numerical results in Table 1, it seems that IFM is comparable to other state-of-the-art methods for image generation (slightly outperformed by PGFM), illustrating that this could be a promising method for data to data transfer.", "weaknesses": "The main weakness of this work is that it seems to be a repacking of the Maximum Mean Discrepancy with a field-induced kernel. Thus, I am unsure of the novelty. Given two probability distributions $p(x), q(x)$ the MMD is the squared distance between their mean embeddings in a reproducing kernel Hilbert space with some kernel $k(x,x’)$. See [1]. From my understanding, in IFM, you are replacing the kernel $k(x,x’)$ with the interaction field (so this is essentially MMD with a field-induced kernel). The IFM loss is then the MMD$^2$. Thus, to me this represents a “repackaged” MMD in the language of fields, and not really a new generative principle. I believe one can derive this mapping to the empirical MMD$^2$ from Eq. 7/the IFM loss. It is essentially comparing distributions via pairwise kernel sums (so is a kernel-based MMD method, e.g. in section 3.2.3). Please correct my understanding if this is wrong, but I think it would be worth addressing the connection to MMD/other distribution alignment techniques.\n\nThe method relies on choosing an interaction field, but there aren’t many details provided about how one should choose this field/in which scenarios it may be beneficial to choose a certain field over others.\n\nThe IFM method performs comparatively well to other methods on image generation tasks but does not outperform significantly. It would be helpful to discuss specific scenarios where IFM could provide advantages over existing methods.\n\nI’m not sure what the point of the image-to-image translation task is besides showing that IFM can do what EFM cannot. It seems relatively simple compared to standard benchmarks in image-to-image translation (e.g. Cityscapes).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wmE0L4Jpjv", "reviewer": "Reviewer_gzz2", "summary": "The authors propose Interaction Field Matching (IFM), a generalization of electrostatic field matching. The authors further design a particular instance of IFM that solves several problems with EFM and show promising performance on several toy and image data transfer problems. The authors further provide extensive theory to show that their model provably transfers between distribution and provide the most general set of requirements for the interaction field to perform data transfer.", "questions": "As someone less familiar with the IFM, I am curious what is the connection between IFM/EFM and flow-matching? Is it possible to generalize them with a single theory?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- The authors provide a strong theoretical contribution by generalizing EFM and providing the most general set of requirements for the interaction field to perform data transfer.\n- The paper is very clearly written, intuitive, and easy to understand\n- The proposed IFM solves several problems with EFM and achieves better performance on several tasks. \n- The proposed IFM is even competitive with/outperforms diffusion/flow-matching methods.\n- Overall, the paper has both very strong theoretical results and promising empirical studies", "weaknesses": "- This is somewhat of a minor point, but the authors only provide experiments on relatively \"toy\" problems. It would be more convincing if the authors included additional experiments on other, more challenging image generation tasks.\n- The paper could use some additional discussion on diffusion/flow matching to situate the work in the broader context of generative modeling", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2zIJt19Ncz", "reviewer": "Reviewer_PXv4", "summary": "This paper proposes a generative model based on EFM, which allows using general interaction fields beyond the existing electrostatic ones. The main theorem shows that IFM can recover the target distribution successfully. Experimental results show that it can outperform competitive models in image generation and image-to-image translation.", "questions": "1. Why does EFM get completely wrong results?\n\n2. Why was this ODE better than a straight-forward linear ODE?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The mathematical results look sound. And it identifies and resolves some problems from previous EFMs.\n\n2. The empirical results are solid, and they are able to demonstrate stronger performance than GANs and DDPMs.", "weaknesses": "1. Ablation studies are not presented in the main paper. It's hard to know what components contribute to the final model gain.\n\nRunning speed isn't provided. So we don't know whether this presented model is, in fact, scalable.\n\nThe resolution of the images used in this paper is quite low.\n\n2. Presentation issues:\n\n(1)  It is better to mark z=L line in Figure 2. It's not very clear now.\n(2)  Equation 11 seems to be incomplete. Or maybe it means an SGD step. It's unclear.\n(3) In Table 1, our->ours.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)