Buckingham $\pi$-Invariant Test‑Time Projection for Robust PDE Surrogate Modeling
Authors: Seokki Lee
Min-Chul Park
Giyong Hong
Changwook Jeong
Venue: ICLR 2026 Poster (2026)
Keywords: Buckingham-pi
PDE
model-agnostic
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We propose a π-invariant test-time projection that aligns test inputs with the training distribution by solving a log-space least squares prob- lem to preserve Buckingham π-invariants.

Abstract: PDE surrogate models such as FNO and PINN struggle to predict solutions across inputs with diverse physical units and scales, limiting their out-of-distribution (OOD) generalization. We propose a $\pi$-invariant test-time projection that aligns test inputs with the training distribution by solving a log-space least squares problem that preserves Buckingham $\pi$-invariants. For PDEs with multidimensional spatial fields, we use geometric representative $\pi$-values to compute distances and project inputs, overcoming degeneracy and singular points that limit prior $\pi$-methods. To accelerate projection, we cluster the training set into K clusters, reducing the complexity from O(MN) to O(KN) for the M training and N test samples. Across wide input scale ranges, tests on 2D thermal conduction and linear elasticity achieve an average MAE reduction up to $\approx 91\\%$ with minimal overhead. This training-free, model-agnostic method is expected to apply to more diverse PDE-based simulations.

Published as a conference paper at ICLR 2026
BUCKINGHAM T-INVARIANT TEST-TIME PROJECTION FOR ROBUST PDE SURROGATE MODELING
Scokki Lee Min-Chul Park Giyong Hong", Changwook Jeong!.9t
1 Graduate School of Semiconductor Materials and Devices Engineering, UNIST
"Computational Science and Engineering Team, Device Solutions, Samsung Electronics
3 AI Graduate School, UNIST
[seokki, e150great, changwook. jeong) @unist. ac. ar.
m. wwwpkaaaasauuuuu com
ABSTRACT
PDE surrogate models such as FNO and PINN struggle to predict solutions across inputs with diverse physical units and scales, limiting their out-of-distributiin (OOD) generalization. We propose a T-invariant test-time projection that aligns test inputs with the training distribution by solving log-space least squares prob- lem that preserves Buckingham I --invariants. For PDEs with multidimensional spatial fields, we use geometric representative r-values to compute distances and project inputs, overcoming degeneracy and singular points that limit prior x- methods. To accelcrate projection, we cluster the training set into clusters, reducing the complexity from ((MN) to O(KN) for the M training and N test samples. Across wide input scale ranges, tests on 2D thermal conduction and linear elasticity achieve MAE reduction of up 91% with minimal overhead. This maining-free, model-agnostic method is expected to apply to more diverse PDE-based simulations.
INTRODUCTION
A central principle of physics is dimensional homogeneity: every valid physical law can be expressed in terms of dimensionless groups, known as Buckingham --invariants. Once recast terms of these invariants, a system's 's behavior remains unchanged under any rescaling of dimensional variables that preserves the values of the v-grrops. For instance, flows in narrow pipe and vast river are dynamically equivalent when their Reynolds number matches. This implies that certain apparent -ut-o--distriuutinn (OOD) shifts, often regarded as a major challenge in physics-int informed machine learning and neural operator models, may not constitute genuine distribution shifts at all, but merely scale changes that are physically equivalent under the Buckingham ı framework.
Building this observation, propose a general, training-frees and model-agnostic test-time procedure, T-invariant projection. Given the input fields and parameters of a partial differential equation (PDE), map each test sample into the neighborhood of the training distribution while preserving its T-values: specifically, we move the sample within its own n-equivalence class and project it onto the nearest training -requivalence class. As result, inputs are aligned according to their essential physical similarity rather than arbitrary scale. The framework is applicable to broad class of PDEs wherever dimensional analysis identifies invariant groups, and integrates inference step for any surrogate model, including U-Nets and neural operators. To reduce compu- tational cost, we replace exhaustive pairwise comparisons with centroid scheme: clustering M training samples into K representatives yields O K N) test-time complexity instead of O(M N) for N test instances. Across diverse physical simulations, --invariant projection substantially improves OOD generalization, often approaching in-distribution accuracy.
drop-in
"qqual contribution. Corresponning author.

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Published as a conference paper at ICLR 2026
2
RELATED WORKS
Fourier Neural Operator (FNO). The Fourier neural operator (FNO) is resolution-agnostic sur- rogate that learns mappings in Fourier space, enabling pooluutioninnependeee predictions (Li al., 2020). Numerous FNO variants followed (Bartolucci et al. 2023; Kossaifi et al.. 2023 Rahman al. 2022 However, truncating to limited set of low-frequency modes can discard physically relevant high-frequency content; Physics-Informn PDE-constrained losses to recover high-free trghufreeyeccy behavior (Li et al. 2024). Overall, FNO excels at in-distribution interpolation but struggles to extrapolate OOD when inputs have disparate units and scales.
Neural Operators (PINO) mitigate this by adding
Dimensionless learning. Learning dimensionless g-groups reduces variables and enforces scale invariance, enabling scalar extrapolation with neural networks or SINDy (Bakarji al. 2022] Xie al. 2022] Oppenheimer et al. [2223] Gunaratnam et al. 2003 Brunton et al. 2016). However, most work focuses on scalars rather than spatial fields. For fields, naive Buckingham scaling has two failures: (i) numerator zeros collapse many inputs to zero; and (ii) near-zero denominators make π undefined or unbounded, destabilizing learning and analysis. These issues are rarely addressed, limiting direct use in 2D/3D. Recent efforts couple gr-rrups with nonlinear maps in turbulent flows (Fukami et al. 2024), but systematic treatment of zero-set collapse and denominator blow-ups remains missing.
Generalization for the Out-of-distribution (OOD) data. Uncertainty methods estimate OOD er-
ror but seldom improve accuracy without fine-tuning (Gal & Ghahramani] 2016 Lakshminaray
2017 Fuchsgruber et al.] 2024) Angelopoulos et al. 2024). Test-time training/adaptation up- dates pretrained model during inference to boost OOD performance (Gandelsman al., [022] Wang et al.] 2020; Adachi et al. 2024), but adds optimization overhead and latency and remains rare for regression and spatial fields. A promising alternative is sample-wise, r--prseervigg test-time alignment without TTT, which is underexplored.
et al.
Our Contributions. This work addresses inefficiency and generalization challenges in PDE-based physical prediction caused by differences in input units and ranges, by introducing a test-time projec- tion based on Buckingham r-invariance. The projection maps test samples near the training samples while preserving a governing dimensionless group, and a centroid tational complexity from ((MN) to OKKN). The module training-free and modell-ggostic, significantly improving OOD performance across various physical simulations, often approaching in-distribution accuracy. The key contributions are three-fold.
scheme reduces compu-
1. x- --ppeserviig test-time projection for spatial fields: an explicit, invariant input transform that maps each test sample toward the training distribution under the governing म, improv- ing robustness to out-of-distribution shifts.
2. r-uniform strategy: uniformizes the sample-wise π distribution by tuning the dominant- scale input while others fixed, enabling balanced training coverage which can be general methodology in dimensionless modeling.
3. centroid reduction for projection: replaces exhaustive pairwise comparisons with centroid representatives, reducing test-time complexity from (MMN to O(KN) with negligible accuracy impact.
Although our proposed framework can be applied to a wide range of PDE analyses, we focused on thermal and stress problems. Additional examples of applying the PDE are shown in App. F In Section 3 we introduce the background of Buckingham-a and define the dimensionless parameters in the thermal conduction and linear elasticity. Section[4] 4| the method for model-agnostic r-invariant projection is explained. addition, at the test-time, efficiett method in terms of computation cost is introduced throughout the T-uniform strategy and K K-means clustering. Section 5 describes the experimental procedure in detail, while Section pressens the corresponding results. Finally, Section 7 present discussion on the experimental limitations and the conclusions.
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3 BUCKINGHAM π THEOREM
The Buckingham x-theorem provides a framework for dimensional analysis, transforming physical variables into dimensionless --groups to reduce complexity and ensure scale invariance, If we know the units of parameters in the given PDE, the g-groups are easily extracted by eliminating the units of denominator and numerator. For system with variables and fundamental units (e.g.. mass, length, time), it yields of these invariants, seemingly distinct inputs with different scales can be recognized as physically equivalent, providing a theoretical foundation for robust OOD generalization. Below, we define representative g-groups for thermal conduction and stress simulation, used motivate invariants and scaling rules.
dimensionless groups (see App By recasting the problem in terms
Theorem 1 ((Buckingham log form)). Given B base units, dimensional variables x E RSo and a dimension matrix D E || (rows = base units, columns variables), let rank(D). Then there exist p independent dimensionless combinations (" ("æ-groupss", If ④b Rex(e-T) spans ker (D), then
log [ogl[(>)= = 4 log
II(x) = exp( + z)
a complete set of independent -rggus.s. (See App. for details.)
Notation.
(6(2) ... cf(-~)) stacks the null-space basis vectors; each column ei defines one dimensionless monomial (a m-group). The set of log-rescalings that preserve all r- is ker( = { 1= Th v== 03.
How scaling acts (log-space translation). A componentwise rescaling x + 3 exp(e) becomes a translation log + z 3 in log space. If u E ker( ), then (= +)+ Φ z, i.e., r-values
are unchanged. Thus each input generates an affine r-equivalence class ker(
In short, the physics lives in the r-values: unit or scale changes are just r-preserving shifts in log space, inputs related by such shifts are the same case.
Worked example (thermal).
With base units ( (M, L, O) and variables (k, q, AT, L).
[
MLT [g] = ML--ML-TT-2
[A]] = O, 14 L.
the dimension matrix can formed.
1
1
0
07
0
D
columns: (k, q. AT, L}, rows: {M, L, T. θ).
(1)
-3
3
0
0 0
0
In general, for a variable with units
II Pb.,, the j-th column of D stores its exponents. The
be B
number of groups defined by the Buckingham æ theorem is rank(D). Because rank D) =3, only one value is defined by the theorem.
A dimensionless monomial corresponds to null- vector ø E ker(D), i.e., Dő = 0. Stacking independent solutions gives r-basis + (0 ... ((p-r)) with DΦ O. but in this case p- = 1, so one vector suffices. A r-basis (null-space basis) is +b = (()) ], where
which produces
9 9I2
Teh = 2 (AT))aA oz = =k-'g'ATT) -ITT'-I'' LI2
k AT'
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(M, L.T, are units of mass, length, time, and temperature, respectively. For the variables, a thermal conductivity field with its unit of [W/mmK]], is a volumetric heat source field with [W/m*], AT is scalar value from the difference between maximum and minimum value of bound- ary conditions with [K], and the grid length with [m]] The Tsh is the constraint our projection enforces (thermal row of equation 12
Worked example (elasticity). Elasticity proceeds analogously. With base units L, and variables (E, J, J, L, Au), the dimension matrix D can be formed.
1
0
0
D == -1 -1 -2
-1
columns: {E, o, f. L, L,Au) rows: {M, L, T}. (2)
2 2 -2
0 0
For the variables, is the Young's modulus field with its unit of [Pa], a stress field with [Pa]. f is body force per volume field with [N/m²], I is the grid length with [m]] and Au displacement value from the difference between maximum and minimum value of boundary conditions with [m].
bbsass (null-space basis) can be collected as
[ (1)))
(
D 0,
with columns
(
+ 31 = o/E,
(x2)
1
IL/E,
1
+
f Au/B
0
0
0
Details are in App. for the task-specific ı and the resulting linear constrnit...
METHOD
We propose training-free, model-agnostic r-invariant projection to align OOD inputs with train- ing data while preserving Buckingham т n-groups (Section B} The pipeline has three stages: (I) r-prrserving test-time projection which projects each test sample toward the training data sample while preserving the æ via dimension reduction, log-scale linearization, projection, and prediction; (2) n-uniform strategy that generate uniform log distribution cover the wide range of group, improving the model prediction accuracy; (3) Centroid reduction for projection: combined with the 7-uniform strategy, we run K-means clustering log-space training features and use the K centroids as projection representatives.
4.1 PROBLEM DEFINITION
We are given training dataset D.r = [Xx,, Y.),, where each input X,, consists of discretized input fields together with the associated physical parameters and discretization step, and each output Y; is the corresponding discretized solution governing PDE. All variables and field values are di- mensional; see Fig. [ This dataset is used to train a chosen surrogate PDE model. Then, for given test input X, our goal is to construct transformed input X that (i) preserves the Buckingham-n metric.
invariants of X and (ii) lies close to the training distribution under a
More formally, let ã = (xx) denote the vector of dimensional variables extracted from X. where feature extractor. For algebraic equations, v/(+) Id(-). For differential equations, we need to extract the representative variables from X using t. which will be discussed later. Then, the problem is defined as
argmin [Zi (M(x.)) []]]
where dist(-) a chosen distance measure between a candidate point and the training set.
'The algorithm of the Buckingham r-invariant projection is shown in App. H
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4.2 DOMAIN PROFILE REDUCTION
For algebraic systems, g-groups can be extracted directly from variables by enforeing dimen- sional homogeneity Theorem I dimensional spatial fields in PDES presents a fundamental challenge. A naive pixel-wise approach often leads to r-degeneracy, as shown in Fig- D if a field value zero (e.g., y -> 0) or near-zero at specific location, the resulting v-value may vanish diverge, causing numerical instability.
However. applying Buckingham-r analysis directly to high-
To overcome this issue, we introduce feature extractor & X ~ that maps each discretized field to finite set of characteristic variables. Among possible choices, we adopt the arithmetic mean of the field values. Unlike pixel-wise evaluation, this global statistic is robust to local zero values and outliers, ensuring non-zero representative scales that enable stable log-linear projection. For non non-spatial inputs, these characteristic variables are used directly.
Thermal
(Length Scale:
Tx i-Information loss (Degeneracy)
Thermal
(Length
Tal
Stress (Length Scale:
Figure 1: The PDE datasets for thermal and linear elasticity. With given distributions of q, f. the surrogate model predicts the fields of ' and a in each case (Thermal and Stress). Pixel-wise π values go zero when - and f - O, which is described r-information loss.
4.3 OFFLINE PREPARATION: T-UNIFORM STRATEGY & CENTROID REDUCTION
The T-uniform strategy is adopted for enabling a balanced training coverage. It uniformizes the sample-wise distribution train samples by tuning the dominant scale input (e.g., q in thermal case) while others fixed. The dominant parameter is defined as the input parameter whose scale most directly controls the Buckingham g-group. Concretely, can be done with SHAP analysis on group for each parameter. For example, the dominant parameter in thermal case shows the highest importance among whole parameter's 's contribution (48.7%). Once the dominant parameter is fixed, first sample the input parameters from broad uniform ranges and compute their r-values. From this n-distribution, we define a target uniform distribution T-space; each training sample adjusts only the dominant parameter (e.g., for thermal). so that its æ-value matches the target while other inputs fixed. By making K clusters in uniformized distribution, the centers of K clusters represent whole distribution. In our approach, the distribution of log æ is uniformized rather than that of π for enlarging the coverage. Naïvely, comparing N tests to all M training samples entails O(M pairwise evaluations of the inter-class residual in equation To reduce this cost, cluster training log-fcatures with K K-means and keep K centroids, cutting complexity to O(KN) while preserving projection accuracy (App. D. Sec. 5.5) (see App.
4.4 ONLINE TEST-TIME PROJECTION
The main goal is to find the optimal projection, i.c., the closest point between the equivalence class generated by the test point X and equivalence classes generated by each train data (X,, Y2). Since the vector that generates equivalence classes in log-space is simply parallel translation, the space it spans is of the form z + ker ( This implies that the entire space decomposes into two parts: (i) the component perpendicular to ker( (inter-class variation, i.e., the physical change). and
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(ii) the component parallel ker( (intra-class change, i.e., the scaling variation). Ultimately, we identify the closest training class to the test class in the inter-class direction, and then adjust the test sample within its own equivalence class achieve scale consistent with the chosen training samples.
More formally, setting 2; llog zt and E log ãt, the optimization problem equation 3 becomes
min dist , (=))m)) min min |2 + v)ll min d_ (1r:), [+], (4)
i= wEker
where indexes the training samples, [ā]x and [ai]s denote the equivalence classes of the and i-th training samples, respectively. Thus, the problem reduces to finding the nearest training class
in terms of the quotient distance between the fixed [+] and the union of the training equivalence
classes [2i See App. B for details on the used metric.
020
Markers
Test Sample z (before) Aligned (after)
A
81
Chosen train
Dg
Arrows
paan
(O MT PanuM inter-class (m-changing)
-0.
M intra-class (1-preserving)
0.3
47
666
1.70
Figure 2: The schematic of m-invariant projection; the test sample (marked as circle) projects to the optimal train sample A:.: D DB).
Step 1. Decomposition of optimal scaling vector
log(2; (z: ( in equation denote the unconstrained log-scaling factor between a, and æ. Each of can be decomposed into intra-class (scaling) and inter-class (physics) parts using the projector onto ker($
Let of 2,
Pl ( 1) TT of
(or =1 --I---000 T with a weight Σ)
PIO
(I PD)
m-preserving)
inter-class (changes
As an example, Figure 2 Jillustrates of and v², which are defined from the test sample म and the two training candidates 1 and x2. For each i, choosing oi Pind yields the quotient distance between [ay] and [ž], the projection aligns the training and test samples in their respective affine spaces, isolating the class-wise irreducible component of the distance (T Pj)ot.
Step 2. Optimum train sample to be projected
The optimum equivalence class be projected has minimum value of ||u of llz among whole training equivalence classes. The optimization problem can be represented as below:
argmin Ioo v2
(6)
where i represents the equivalence class number to be projected. For example, in the Figure . the optimum point because DA = ||o| - ville < DB = llug - vz 12/12. After that, the optimally
scaled test sample is finally given by
= exp(2") = exp(2 + v²) = I @ exp(x").

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Physical Input (Thermal)
Domain Profile Reduction
Multi-Chmmen Channel Input (ka de. dy)
c-mmwrint projection ts R. 6
Ropresentative log.
(k. F,
Shater Ka,Naa,
K (ay)
Q-167
x-imarand scaling rule 02 (233660)
y, 8)
Projected Input 0'.e' aar"" (ty)
ostest-strail
(=k)oT))
log Bi
subgeet
Neural Network
Physical Space
(R.q. T.P.
(in pu. (ou?)
Dimensionics
Log Space
Linearize
Physical Output
comfour (Tes))
Buckingham -Invariant Projection
Predict
Figure 3: Flowehart of Buckingham x r-Invariant Projection (Thermal case).
EXPERIMENTAL SETUP
The experiment proceeds in the following order: Training (Sec. 5.1 Domain Profile Reduction (Sec. 5.2 -> -iivvariant Projection (Sec. (53) - Prediction (Sec. 5.4 to perform OOD data prediction (Figure 3). In addition, for more efficient --invariant projection, the æ-uniform strategy and centroid reduction method can also employed (Section 5.5)
5.1 TRAINING
Thermal conduction model We solve the steady 2D conduction problem with Dirichlet boundary conditions on The:
(AVT) q-
We use three surrogate architectures: a plain CNN, U-Net, and FNO. Training and test sets are generated by numerically solving equation 7 on uniform grids, but with disjoint parameter ranges induce OOD shift. For example, log1o 9 E [0, 7.5] for training and log q [7.5, 12) for testing. Distributions of k, q. The, and dr are shown in Figure (thermal panels). we encode this input into five channels (ke, q, be: dx, dy).
Linear elasticity model
We consider plane stress/strain in 2D:
(8) As in the thermal case, we evaluate CNN. U-Net, and FNO surrogates. Inputs are field maps E, v, and f, the displacement boundary condition the. and grid spacings dr, dy. We encode f and Ube via components (faz fys uhe ug.)) yielding eight input channels: (E, fy. ube ube, v, dæ, dy). Parameter distributions for (E, f. Ubes da)) are shown in Figure 4 (elasticity panels).
-V -v a= =CC(Evv) : =(u).
5.2 DOMAIN PROFILE REDUCTION
To represent global characteristics of the domain, arithmetic mean values of input fields are adopted.
As defined in Section 4.2 fields of k, q (thermal) and E, (elasticity) are summarized to represen- tative values k. g. E, respectively. However, representive f (f) may be extracted in a slightly different manner, because it is represented two channels corresponding to fa, fy components. Therefore, in this case we first compute the means fr. fy of fa. fy, respectively, and then compute
using the Lo-norms of fa, fy (f v f²² + F2y). we set dr = dy = unless otherwise stated. For quantities that have zero values are not considered. The resulting feature vectors are shown in the pipeline (Figure

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5.3 TTINNARIANT PROJECTION
Prior to projection, the scaling process essential in projection, enabling the surrogate model to relocate OOD onto interpolative (in-distribution) scale and reliable predictions. For the thermal case, the scaling coefficients at [3, and are used to scale the parameters while preserving the Thh Log-scale linearization is applied for efficient calculation (Section 3
K = oki o' 3dj AT" >AT, =1
(9)
For preserving the Teh scaling coefficients should follow the constraint.
d'L2 (8L) = KAT Thh 1 AT (ak)(7AT)
OL²
(10)
Likewise, scaling coefficients are applied each input in stress case as below:
E'
aE, =E= I =3f; L' ='
L'L' (85(G)) IL (a
(11)
Wel
E
E
Log formats are used to equations
25
linearly transform scales of parameters.
log 3 + 2 log - log log 0 (thermal)
(12)
log B + log log a (elasticity)
After that, inputs [k: g, AT, L] of given test sample v are scaled by exp(e") =[a, 3. and moved to the optimal projection point 3 having inputs x* exp(e") = jak, 3q.
which preserves Tth Fel in Appendix A.1
YAT, 8L] from calculating the nearest class with x presserring
correction described in Section 4.4
The same rule is applied in the calculation of the elasticity.
(Poisson generation)
Elasticity (linear, plane
Proebally Demsity
(+)
Densty
(h)
logio (4)
logio (q)
logio (E)
logie (f)
Prosbblty Denaty
(c)
(d)
(9)
())
Proablity
500
0.011
0.012
0.0
0.5
1.0
0.010
0.012
0.014
Tae
dx
dk
Prostaiiy Dersty
(o)
(K)
(Befors)
(ABtrr
(Beffoor)
(ABte))
distribtion
Figure 4: Histogram comparison for train (blue) and test (orange). Thermal (a-f): (a) logın k, (b)
log10 9. (c) The, (d) dr; (e) log1o Teh before and (f) after -uunffrrm sampling. Elasticity (g-l): (g)
log10 F., (h) log1o [] (i) the. (i) dr, (k,1) log1o Tel (before vs. after).
5.4 PREDICTION PROCESS (SCALING, INVERSE SCALING)
After computing the optimal v*, we rescale the test input channel-wise X - X exp(u"),
X [ack, 3q, Tie,, ōdx, ödy] (thermal)
(13)
[a ffz, Bfy, yuh, yuk v, 5dz, ödyl (elasticity)
Feeding X to the surrogate model predicts Tscatess and invert the temperature scaling using the AT scale factor y: Tmin + + (Teated Tmin). where Tmin is the minimum Dirichlet boundary temperature. (Analogous inverse maps apply for elasticity outputs.)
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5.5 CENTROID REDUCTION FOR PROJECTION
In calculation the nearest class in Section 5.3 We apply a n-uniform sampling strategy (Sec 4.3 to
obtain an approximately uniform distribution of log x (Figure 4 (e,f, (k.l)). Prior to applying the strategy, the log distribution is one-sided and skewed; afterwards it becomes approximately uni- form, indicating that the training space covers a broader range of ग values. For thermal we vary q to target desired Tth values; for elasticity we analogously vary f for Tel- After æ-uniform sampling, we run K-means on the training log-features 2 obtain K E (1..... 10) centroids. Because K K-means is stochastic, we average results over 10 seeds (for both clustering and test sets). We compare Base- line (nearest over all training samples), Clustered (nearest over K centroids), and Random (nearest over K random training samples) using MAE, RMSE, and wall-clock. Timing excludes surrogate forward passes and reports only the projection stage (candidate selection + solving equation 3
6 RESULTS AND DISCUSSION (THERMAL AND ELASTICITY)
Figure 3 compares Top-3 best and Top-3 worst OOD cases for thermal and elasticity. Across both, л n-projection substantially improves fidelity, with the largest gains the worst cases where the raw surrogate fails to extrapolate. Table T reports MAE/MSE and projection time for each architectur With the Buckingham-T projection, both thermal and stress simulations achieve lower MAE and RMSE errors, but the inference time increases. However, applying the cluster algorithm with sig- nificantly reduce the inference time, while achieving a level of accuracy comparable to that obtained when computing distances over all training data.
The centroid reduction algorithm as described in 5.3 has demonstrated the efficiency in terms of computation cost when predicting the temperature distribution without a significant degradation in accuracy. From Figure [6] it is shown that the MAE tends to converge with the baseline projection as the number of candidates (cluster centers and random samples) increases while making gap with random projection. The clustered projection matches Baseline MAE within statistical noise while reducing projection time by ~ 100x (Figure
Handling n-degeneracy. When Tth is ill-defined or ill-conditioned (e.g., R 0 or AT Rs 0), we drop the explicit constraint and solve the same log-space alignment on the non-degenerate chan- nels (c.g., k, AT, dæ, dy in thermal). This removes global scale mismatch while preserving spatial heterogeneity, which empirically restores OOD accuracy. When valid g-groups are available (e.g., advection -itettooidiffuiionn. Navier-Stokes, the same log-affine machinery applies with the corresponding constraint.
Table 1: Test scores on thermal and elasticity. The best scores are bolded.
Thermal RMSE
Stress RMSE
Method
MAB
Time
MAE
Time
CNN
8.43+-0.29
9.99--033
0.96+-008
1.17400.10
CNN Paiewise Projection
2.6310..04
1.24.10.05
100,3112.06
0.53:10.0M
0.71:10.08
13.58.10.59
CNN r-unifomn Rnnnansoms
1.85+-0.14
2.30+0.11
1.75:1.0.02
0.62++0.05
0.86+0008
1.31:0.044
CNN + r-unifoom 10 Centroids
1.79:10.08
2.23:1:0009
1.80.1.02
0.60.10.05
0.84
1.3610..4
U-Net
13.60=0.331
15.29--0.35
0.81+-0.08
0.99-00.0
N.Net Pairwise Projection
1.75
2.31 +0002
99.12-1..0
0.17:1:0.03
0.28:1.000
94.91 1.70
U-Net+ 7-uniform Randoms
1.2010..11
1.5610.113
2.25-1:0.08
0.23:10003
0.4010.00
0.50
U-Net n-uniform 10-Centroids
1.18:10.08
1.53::0.110
2.31:0007
0.22::003
0.39:20.88
1.58=0.07
FNO
9.88
11.43.0.26
3.201.0.84
4.19
FNO Pairwise Projection
1.38+0.02
1.74+0.02
151.4443.1 16
0.28:1-0.03
0.42-20.08
94.02+1.10
FNO u-uniform- 10. Randons
1.29::0.12
1.65
2.24:1..04
0.34100.03
0.04
0.45
ENO uniliomm 10- TU-Centroids
1.25::0.10
1.60:1:0.11
2.3110006
0.33110.03
0.53110.08
1.54.10.04
LIMITATIONS AND CONCLUSION
I Limitations. Our m-invariant projection assumes PDE-govermed behavior and may underperform in hybrid settings with empirical components. Using representative statistics (c.g., arithmetic means
We follow the authors' public implementation and adapt image-wise outputs; details App.

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Published as a conference paper at ICLR 2026
(a) Top-3 best Cases (Thermal)
(b) Top-3 worst Cases (Thermal)
Tasee
Tored (projected)
Tava (bsseline
Time
Tyxo (propetle)
Tyws (baseline)
.40
4D
440
-20
-30
10
(c) Top-3 best Cases (Stress)
(d) Top-3 worst Cases (Stress)
(orojacted)
(basaline)
Mgue
Moved (projacted)
Ujred (basaline)
1.5
15
13
10
10
0.
08
23
10
10
10
75
C25
D29
320
Figure 5: Comparison of true and predicted fields for Top-3 best (a) and Top-3 worst (b) cases in thermal (temperature) and elasticity (stress), under raw input vs. Buckingham --invariant projection.
(c)
Randon profection Beceline projection
Clustiing projecton
40
Clucsering p siecdon Randow projectonn Basseline projection
C untering projection Randon projection
Booeline projection
10
MAE
RMSSE
10
10
15-
10
3
1102
3
1000
5
1000
Namber of candidates
Number of candidates
Number of candidates
Figure 6: Performance comparison on (a) MAE, (b) RMSE, and (c) Time cost between cluster- ingrrandom/baseline projection methods with 10 different test sets.
k. ②) can blur highly irregular spatial patterns. While 7-uniform sampling widens scale coverage, extreme OOD beyond the trained स range still degrades accuracy (see App. G
Conclusion. We introduce a compact, training-free and model-agnostic --invariant test-time projection that enforces physical similarity via Buckingham-r and aligns OOD inputs the training manifold through tiny log-space least-squares scaling. This explicit log-scale transforma- tion enables accurate predictions even under extreme out-of-distribution shifts. A centroid-based clustering variant accelerates inference, reducing complexity from O(MN) to O(KN) with negligible accuracy loss, and outperforms traditional test-time training in speed while maintaining high performance. On 2D thermal conduction and linear elasticity. the method reduces MAE by up to zs 91% with minimal overhead. Future work includes uncertainty- projections, extensions to transient and convective PDEs (c.g., advection diffusion.. Stoikr-Stokes), and hybrid settings that incorporate empirical components.
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ACKNOWLEDGMENTS
Changwook Jeong acknowledges support from the Nano Material Technology Development Pro- gram through the National Research Foundation of Korea (NRF), funded by the Ministry of Science and ICT (MSIT) (2710089547); from the Institute for Information Communications Technology Planning Evaluation (IITP) grant funded by MSIT, Korea (RS-2020-11201336. AIGS); and from Samsung Electronics 00250068-1309743
REFERENCES
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Anastasios Angelopoulos, Rina Foygel Barber, and Stephen Bates. Theoretical foundations of
conformal prediction. arXiv preprint arXiv:241.. 1./22411.1888 1..122411../884, 2024.
Joseph Bakarji, Jared Callaham, Steven L Brunton, and Nathan Kutz. Dimensionally consistent learning with buckingham pi. Nature Computational Science, 2022.
Francesca Bartolucci, Emmanuel de Bezenac, Bogdan Raonic, Roberto Molinaro, Siddhartha Mishra, and Rima Alaifari. Representation equivalent neural operators: a framework for alias-free operator learning. Advances in Neural Information Processing Systems, 36:69661-69672, 2023.
Steven L Brunton, Joshua L Proctor, and Nathan Kutz. Discovering governing equations from data by sparse identification nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 2016
Dominik Fuchsgruber, Tom Wollschläger, and Stephan Günnemann. Energy-based epistemic un- certainty for graph neural networks. Advances in Neural Information Processing Systems, 37: 2024.
Kai Fukami, Susumu Goto, and Kunihiko Taira. Data-driven nonlinear turbulent flow scaling with buckingham pi variables. Journal of Fluid Mechanics, 2024.
Yarin Gal and Zoubin Ghahramani. Dropout as a bayesian approximation: Representing model uncertainty in deep learning. In international conference on machine learning, pp. 1050-1059.
PMLR, 2016.
Yossi Gandelsman, Yu Sun, Xinlei Chen, and Alexei Efros. Test-time training with masked autoen- coders. Advances in Neural Information Processing Systems, 35:29374-29385, 2022.
David Gunaratnam, Taivas Degroff, and John S Gero. Improving neural network models of physi- cal systems through dimensional analysis. Applied Soft Computing, 2(4):283-296. 2003.
Jean Kossaifi, Nikola Kovachki, Kamyar Azizzadenesheli, and Anima Anandkumar. Multi-grid tensorized fourier neural operator for high-resolution pdes. arXiv preprint 10X0v:2300.00100 2023.
Balaji Lakshminarayanan. Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. Advances in neural information processing systems, 30, 2017.
Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, An- drew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08899 2020.
Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli, and Anima Anandkumar. Physics-informed neural operator for learning partial differential equations. ACM/IMS Journal of Data Science, 1(3):1-27, 2024.
Michael W Oppenheimer, David Doman, and Justin Merrick. Multi-scale physics-informe machine learning using the buckingham pi theorem. Journal of Computational Physics, 2023.
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Md Ashiqur Rahman, Zachary E Ross, and Kamyar Azizzadeneshelii U-no: U-shaped neural oper- ators. arXiv preprint arXiv:2204.11127, 2022.
Dequan Wang, Evan Shelhamer, Shaoteng Liu, Bruno Olshausen, and Trevor Darrell. Tent: Fully
test-time adaptation by entropy minimization. arXiv preprint arXiv:2006. 2020.
Xiaoyu Xie, Arash Samaei, Jiachen Guo, Wing Kam Liu, and Zhengtao Gan. Data-driven discovery of dimensionless numbers and governing laws from scarce measurements. Nature communica- tions, 2022.
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A DIMENSIONAL ANALYSIS VIA MATRIX NULL SPACE
By following the processes for each task, Buckingham т groups can be defined.
A.1 THERMAL CONDUCTION
Let the base dimensions be B {M, L, T, } (i.e. mass M. length L, time T, and temperature
)). We encode units as exponent vectors x over В. For example, the unit of thermal conductivity
(k) [W/ K)] is expressed as Fy [ -' ']. and taking transpose, it is transformed into 2/ = [1, 33 -1]". Stacking the columns F& 2y TAT,I for the variables [k, q. AT, L] forms the dimension matrix D R xp where p a total number of input and output parameters. The units and parameters are (thermal conductivity, [W/(m K)D, q (volumetric heat source, (W/m*D. (grid length, [m]),
and 7 (temperature, [K].
1
0
07
-1
0
D
columns: {k: q. AT, L}, rows: {M, L, T, θ}. (14)
3
3
0
0
-1 0
1
0J
In general, for a variable with units
I1 Pob, the j-th column of D stores its exponents. The
be
number of groups defined by the Buckingham π theorem is rank(D). Because rank =3, only one value is defined by the theorem.
A dimensionless monomial corresponds to null- vector ø E ker(D), i.e., Dø = 0. Stacking independent solutions gives r-bbsis & [o(), ... Ø(p-r)] with DΦ O, but in this case p = 1, so one vector suffices. A x-basis (null-space basis) is & (! }. where
which produces
g L
Tth 1: u0 (AT) 1, = (
KAT
Unit check:
(W/m²) (m²)
W/m
=1
(W/(m K)) (K)
W W/m
so Tth is dimensionless (any nonzero scalar multiple of o yields the same π up to constant power).
A.2 LINEAR ELASTICITY
We use base dimensions B = (M, L T) (no temperature). Let the variables be (E, a. f. L, Au)
with units [E] = [o] = M 1T- [f] M L- 2 and [L] = [Au] = L. The units and parameters are: E (Young's modulus, [Pa]). f (body force per volume, (N/m)), (grid length, [m]], u (displace- ment, [m]), and (stress. [Pa]).
The corresponding dimension matrix (rows M, L, T; columns as ordered) is
1
0 07
D=
- -1 -2
-2
-2 -2 0 0
Here p 5 and rank(D) = 2, the nullity is p rank(D) D) 3, i.e., three independent r-groups.
b--bsii (null-space basis) is collected as
[ (1) (
(3)
DΦ 0,
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with columns
((1)
+ æ1 = o/E,
((2)
1
122 = IL/E,
2(3)
7 T3 Au/E.
0
(If grid spacings are anisotropic, replace L by the arithmetic mean Left vdx dy; equivalently, use (log dæ + log dy) in the log constraints.)
A.2.1 MULTI T-INVARIANT PROJECTION
At test-time we enforce only input-side x-groups avoid using unknown outputs. Thus we exclude T o/E and enforce T2 = E; ; if Au is provided as an input, we may additionally enforce 33 = /u/ E for tighter projection. Default: Au unavailable, enforce only π22 = f F.
Let the input-side log-scales be [log a, log 8, log log 8] acting on E, f: L, Au respectively, i.e.
E' = aE,
= L' =L=
Au' Av.
Then the enforced T-constraints become linear equations in v:
1
log B + log y log (m2 (e::[L/E) log 3 + log log a 0 (TT fuuu E).
1 v v=0 =
C
Given test log-features: v and candidate train 2;, define of =. Project of onto the r-preserving subspace using either
ii projector onto ker(C): P PI=1-CT
= ='= P n,i
Or equivalently
(ii) KKT solve for min v -
112 s.t. Cv = 0.
Step 1 (nearest class under multiple r')).
arg min (1 - P1) P1)
(15)
Step (r-preserving correction). Apply Pi oft and rescale channels by (a, B, 7, 8) exp(v*) obtain the projected input. Run the surrogate on the projected input and inverse-scale the outputs (details in App. B).
Notes. (i) If only one of {L, Au) is present, the system reduces to the single-constraint case (en- force fL/E fāu/E) E).
(ii) Under anisotropic discretization, replace the L-row in C by " log dr + log dy (i.e., use Leff /dr
(iii constraint becomes ill-conditioned (e.g., f R 0), drop that of C and project with the remaining valid constraints (degeneracy fallback).
B INPUT-SPACE L2 METRIC ON LOG-SCALE
Definition (Logarithmic distance). For x. y E Rh with logarithmic coordinates = log x and log y, define the distance
d(x,w) v) ||loog - log wl12 ||= w|12-
Then, a metric on R. because x + log injective on x E RS, and [|:112 is the standard Fuclidean norm.
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Before n-uniform
After n-uniform
Before IT-uniform
After n--uniform
Train point
contours (Train)
Figure 7: The schematic of 7-uniform strategy
Remark 1. Under the multiplicative scaling action as in Section
we have
d(p(u, ((..,)).ii p(u. v)) [1(=++(]] + 1) ||> w|l2 d(x,,)) v),
Hence, the metric d is invariant under this group action.
Remark 2. The metric induces the standard point-to-set distance from a point log y to the n-equivalence class [z)] (or equivalently [z]= = {()) of fixed log x as
dist(y, [x]]) = dist (w, {(=))
inf veker TT ||w (z + v)
Then, define the class-to-clase (quotient) distance by
dist(ly)a: [z]m] dist(E(w),E(z))
inf [[( u) (2 + u2 a.veker
Since d invariant under translations by -u E kcr(# T), this reduces to the point-to- distance:
dist ((w)), E(z))
inf ||| 1-)l22) inf || w - (= + 8) || = dist(w, {()).
u,veker T
sEker T
C
T-UNIFORM STRATEGY
Figure 7 shows the schematic of z-uniform strategy. The dense r-contours train sample are distributed uniformly by tuning dominant-scale input parameters (e.g., 7. f in thermal and elasticity case, respectively) so that the set of centroids can cover the distribution of all train samples when clustering is performed.
D CLUSTERING PROJECTION
The clustering projection proceeds as follows: (1) Uniformly adjust the pi distribution of the train samples using the T-uniform strategy. (2) Perform K-means clustering based on the π values and determine the centroid for each cluster (marked with star symbol). (3) Find the closest centroid for a given test sample and project the test sample to the closest position on the Trest plane (T inviiiarinnt projection).
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Cluster
Cluster
Cluster 2
Center 0
Center 1
Center 2
Thass plane
1.5
Test sample
Test sample (projected)
mT plane
1.0
0.5
Equivariant classes
IT plane
0.0
P ( u
To plane
-1.0
Scaling change
-1.5
1.96
0.8
2.00
1og
0.6
10
4 T
0.4
2.04
109
P T))
0.2
2.06
Physical variation
Figure 8: The scheme of clustering projection. Train samples within the same cluster are represented the same color with centers. The test sample projected to the nearest center of the cluster while preserving its Tresti it moves on the Teest plane.
E EXPERIMENT METRICS
E. MAE AND RMSE EVALUATION
In this experiment, image-wise metric is used for evaluating the MAE, RMSE. For the each image matrix (Yig), MAE can be calculated by
H W
1
ΣΣ
(()
vr(n)
MAE,
(16)
H W
==
Here, H and W denote the image's height and width, respectively. (in this experiment image spond to color map). using equation T6 image-wise average MAE can be calculated by
MAE
MAE,
(17)
N
Additionally, RMSE can be calculated by
H
W
RMSE,
(n)
mm))
(18)
H
W
from equation 17 image-wise average RMSE can be calculated by
RMSE 1 RMSF,,
(19)
N
F ADDITIONAL PDE EXMAPLE: NAVIER-STOKES EQUATION
We did experiments on Navier-Stokes equation and proved that the projection method also holds for nonlinear problems. The steady governing equations
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pu Vju = -Vp+ + ff in Ω vuu= in 2
(20)
(21)
where u = (Uz, My) is the velocity, p is the pressure, and are density and dynamic viscosity, and f is a body force. We consider a domain 52 with Dirichlet velocity boundary conditions on 8hp. The kinematic viscosity is V píp. and the Reynolds number is Re = PAUL/H where
Ubc
AU is the difference between the maximum and minimum value of U V 2 + u on boundaries.
This Reynolds number (Re) is used for dimensionless number of the g-gropp. We declare the convergence to steady state when the quantity, ||u =f,l22 + |lul uy 2| (implemented as
the sum of the & norms of Ur and By where At 10-1) drops below a tolerance of 10-6
F.1 LAVIER-STOKER EQUATION (AN IDEAL CASE UNAFFECTED BY EXTERNAL FORCES)
To represent the ideal case, f is set to O, assuming that external forcing such as gravity and flow is not present. In this case, the steady governing equations are
p(u V VV in Ω V u = in 2
(22)
(23)
For the model training, input data is encoded with 6 channels X =[p, H4, u u dr, dy), output data encoded with channels (uz uy]- The scaling coefficients a, 3,7 and 5 are used to scale the parameters while preserving the Re.
ap, ='= AU' = ~UU L' ='=
(24)
For preserving the Re, scaling coefficients should follow the constraint.
D'L'AU" (up)(8L)(AUU) pLAU (3p)
Re
(25)
fz'
For efficient calculation, log-scale linearization applied. After the r-invariant projection, we rescale the test input channel-wise X X @ exp(x")) Then
= =ap 31,, yubc: yu%e 5dz, 5dy]
(26)
By using the AU scale factor 7, output u = (uz; uy) can be predicted. We observe that applying the T-invariant projection the OOD data yields higher performance than making predictions without the projection (see Table 2). Top-3 best and Top-3 worst OOD cases for Navier-Stokes are shown
in Figure
Table Test scores on Navier-Stokes. The best scores are bolded.
Method
MAE
RMSE
CNN
1.063.0.000
0.100.10.00
CNN + Pairwise Projection
0.016:10.000
0.022-10.01
27.97-0099
CNN + r-uniform 10-Randoms
0.018+t0.005
0.024+0.001
0.91-10.03
CNN T-uniform 10-Centroids
0.017#0.001
0.023=0.001
0.97+0.02
U-Net
0.11110007
0.15:10.01
U-Net + Pairwise Projection
0.02-1:0001
0.02-1:.001
08.23++0.76
U-Net 7-uniform 10-Randoms
0.03.10.02
0.037.10.02
0,92-1:002
U-Net + 7-uniform 10-Centroids
0.02:1:0.01
0.03++0.001
0.98+00.22
FNO
0.034
0.044
FNO Pairwise Projection
0.013-10.001
0.018100001
28.49=00777
FNO r-uniform 10-Randoms
0.016:10.001
0.02110.001
1.01--0.04
RNO r-uniform 10-Centroids
0.014++0.001
0.019+0.001
1.07++0.05
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Top-3 best Cases (M₂)
Top-3 worst Cases (u,)
hunl
5.3
140
451
641
ard (1
prod 1961
0.4
and 11 11
prod mrne
on
4405
Top-3 best Cases (uy)
Top-3 Cases (u,)
600
Figure 9: Comparison of ground truths and predictions with Top-3 best (a) and Top-3 worst (b) cases in Navier-Stokes, under baseline (raw input) vs. Buckingham r--ivvariatt projection.
F.2 NAVIER- STOKES EQUATION (AFFECTED BY EXTERNAL FORCES)
The ideal form of the Ntvier-Stokes equation does not include gravity or any externally imposed flow (f 0), additional considerations are required when external forcing present (f * 0). However, the π- groups used in the Buckingham -T projection are derived under this ideal assumption (f (==), 0). Therefore, when +f O, the resulting т-groups become incomplete.
We also performed experiments to examine how this incompleteness affects the results when such incomplete x- x-groups are used. In this setup,
8pU
(27)
pT?
used, where ju is the viscosity, is the boundary-condition velocity, L is the characteristic length, and p is the density. The characteristic length L was computed the product of the grid spacing (Arr) and the resolution.
Table 3: Test Score Comparison for Navier-Stokes (f O vs. f = 0) with 10 different test sets. The best scores are bolded.
Method
Navier-stokes (f
stokes RMSE
MAE
RMSE
Time
MAE
Time
CNN
0.063 + 0.008
0.085 1: 0.010
0.063 0.004
0.100 = 0.007
CNN Pairwise Projection
+ 0.002
0.037
0.003
28.01
0.016
0.001
0.022
27.97 -
U-Net
0.105 + 0.003
0.141
0.006
0.11 1: 0.007
0.15 : 0.01
U-Net Pairwise Projection
0.061 1 0.002
0.081
0.002
28.19 0.80
0.02
0.001
0.02 0.001
28.23 0.76
+ 0.007
0.044
0.003
FNO Pairwise Projection
0.025 + 0.002
0.032 0.002
28.56 + 0.73
0.013 0.001
0.018 + 0.001
28.49 0.77
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Table shows that even in the incomplete case (f * 0), applying the Buckingham- projection improves prediction accuracy. Although the accuracy slightly lower compared to the ideal case (f 0), this result demonstrates that the Buckingham-r projection can still enhance prediction performance under incomplete conditions.
G OUT-OF DISTRIBUTION DATA IN DIMENSIONLESS SPACE
In this experiment, we evaluate how much the improvement offered by the Buckingham-n invariant projection degrades when the dimensionless numbers extracted from the imputs of OOD data lic outside the r-distribution obtained from the training data, depending on how far they deviate. For the experiment,
the Reynolds number (Rc)-one of the r-groups commonly appearing in the Navier-Stoker Stokes equations- as the domain variable. This choice motivated by the fact that the flow behavior changes significantly with the Reynolds number (Re), making suitable quantity for explaining and interpreting the results of our experiment.
For example, when the Re is below the flow is dominated by viscous effects and typically exhibits Stokes flow. When the Re is in the range of 10-40, attached vortices commonly appear, while for Re greater than 40, the flow dominated by the Von Kármán vortex street. In our setup, the model is trained only on data with Re in the range 0.04-1.01 (Train data), and tested on three datasets:
Test-ID 00.04-1.02 Test-OOD (mid) with range of 0.32 27. 27.06, and Test-OOD (extreme) with range of 42.28 63 43 (shown in Figure 10
Train
Probability density
Test-ID range)
Test-OOD (mid) Test-OOD (extreme)
1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Rog
Figure 10: The distribution of log Re. Train Re range: 0.04-1.01, Test-ID Re range: 0.04-1.02,
Test-OOD (mid) Re range: 0.32-27.06, Test-ID (extreme) Re range: 42.28-64.43.
Table 4: Test scores on Navier-Stokes with 3 different test sets. The best scores are bolded.
Test-ID (in-range) RMSE
Test-OOD (mid)
Test-OOD (extreme)
Method
MAE
R2
MAE
RMSE
R2
MAE
RMSE
R2
CNN
0.02
0.03
0.02
0.20
0.28
0.12
0.38
0.59
-0.04
CNN + Pairwise Projection
0.005
0.009
0.82
0.08
0.13
0.76
0.34
0.52
0.18
U-Net
0.01
0.02
0.50
0.23
0.30
0.10
0.38
0.54
0.18
U-Net Pairwise Projection
0.003
0.004
0.97
0.06
0.09
0.92
0.24
0.32
0.70
FNO
0.006
0.95
0.20
0.26
0.39
0.33
0.45
0.14
FNO Pairwise Projection
0.002
0.002
0.99
0.04
0.05
0.97
0.21
0.27
0.78
Table 4 shows that as the test dataset moves farther away from the training dataset, the prediction accuracy gradually decreases despite the improvements provided by the m-invariant projection. For the CNN model, the Test-ID whits similar to the training distribution- achievess an R² score of 82%. However, the score drops to 76% for the Test-OOD (mid) dataset, and further down to 18% for the Test-OOD (extreme) dataset. This trend appears consistently across different models, including U-Net and FNO. Nevertheless, interesting observation from these results is
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that even in cases where the data lie far outside the x-distribution, using the r-invariant projection still improves prediction performance compared to making predictions without Although the degree of improvement diminishes when the test dataset deviates too far from the training data, the r-uniform strategy proposed in the paper provides a way to uniformly cover this range. Moreover, in practical settings, the physically meaningful ærrange can be predefined based on prior knowledge from physics and engineering.
H ALGORITHM
Algorithm Buckingham--rinnariiin test-time projection
1: Train surrogate fo Dtrain {(X,, Yi)
2: for each test pair (X,, Y:)= E Drest do
OOD test set, Drest {(x, )=
v- extractor 4: Set - 2; 2 and decompose v into Pipd (intra-class) and (I P1 )uf (inter-class).
2 log (X)
5: Select the nearest train sample by
6: arg mini; |of of la where vg = Plu
7: Form projected input X X o exp(u;.): ): and predict Ysraled f- fo(
8: Yorig f INVERSESCALL scaaled, v;.))
b e.g., Tiin + 7 (Tscaled - -Tmin):
9:
end for
20


Reviews:
[{"review_id": "by6lq9bqXN", "reviewer": "Reviewer_VJx6", "summary": "The paper proposes a projection method, called \\pi-invariant test-time projection, which can cluster training and test data points into to dimensionless groups. Such group is invariant to the unit scales, and can help separate data points with different physical behaviors.  According to the paper, this can help reduce the prediction performance for OOD samples.", "questions": "see above", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The introduction of Buckingham \\pi-invariant is interesting and new to the CS/ML community --- though this might be an old concept in computational physics/applied math. \n2. The idea of clustering data points according to different behaviors without being influenced by units/scale changes is interesting and has pontential to enhance the generalization of current ML surrogate models.", "weaknesses": "1. Clarity. This might be the biggest issue --- the paper does not explains clearly how the proposed projection method is integrated into the training/testing pipeline of neural operators to enhance OOD prediction. Throughout, the paper focuses on how to do projection and clustering. but what to do with NO training/testing? Will you train a different NO for each cluster, and then use the test-time projection for each test example to dynamically determine which NO should be used to predict? Or will you compute a soft cluster membership of each cluster, and then perform a mixture of predictions? Intuitively, there can be many ways of integrating the proposed method with NO training/testing or even data preparation/acquisition. However, the part is significantly lacking and it is hard to understand how the improvement in experimental part is obtained. \n\n2. The OOD problem mentioned in this paper is a bit different from commonly used settings. Regarding OOD, we will first change the distribution of the input to neural operators, rather than assume fundamental change of physical behaviors. In fact, an ML surrogate (e.g., NO) is typically used to capture one type of physical behaviors under various scenarios. I am not sure if expanding the scope to make a surrogate model account for several different physics is appropriate or feasible. At least, the paper should clarify its own meaning of OOD, difference/connection with settings in prior works. \n\n3. The experimental results are limited. Only on two systems is not sufficient in this community. Also, there is no standard deviation in Table 1, making it hard to conclude the significance of improvement.", "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": "fWvz3kcWT0", "reviewer": "Reviewer_pqY4", "summary": "This paper focuses on mitigating out-of-distribution (OoD) inference of neural operators. The method is based on Buckingham $\\pi$-theorem, which decomposes the entire spaces into two parts: the null space $\\ker(\\Phi^T)$ and the component perpendicular to $\\ker(\\Phi^T)$, where $\\Phi=[\\phi^{(1)}\\cdots\\phi^{(p-r)}]$ stacks the null-space bases vectors (see Theorem 1). Thus, data can be transformed or projected to a point that is the closest to training data while preserving $\\pi$. Experiments demonstrate effectiveness on OoD data that is superior than baselines.", "questions": "See weakness.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "**Originality** is high. The method leverages Buckingham $\\pi$-theorem and addresses OoD problem innovatively from a structured perspective, i.e., data space can be decomposed into equivalence classes generated by training data $(X_i, Y_i)$.\n\n**Clarity** is good in general except for some minor issue (see weakness). Diagram and figures are illustrative and helpful.", "weaknesses": "Regarding **Clarity**:\n1. An algorithm that summarizes all the procedure can be helpful to readers. Especially, corresponding to predict & inverse in Fig. 3. How to do inverse in general?\n2. Above equation (11), should $\\tilde{X}^{\\star}$ be $\\tilde{X}^{*}$ as eq. (11)?\n3. In Fig. 1, what do purple circles and yellow circles stand for?\n\nReference:\n1. How is your work related to Lie point symmetry [1]?\n\nLimitation:\n1. Can your method be applied to irregular mesh grids? All baseline models in your paper, CNN, U-Net and FNO, can only be applied on uniform grids. Is there such limitation for your method?\n\n[1] Brandstetter, J., Welling, M., & Worrall, D. E. (2022, June). Lie point symmetry data augmentation for neural pde solvers. In International Conference on Machine Learning (pp. 2241-2256). PMLR.", "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": "s0M68G9GSV", "reviewer": "Reviewer_QjXX", "summary": "This paper proposes the use of a test-time projection method in which samples are rescaled to match the scaling of the nearest point in the training data without changing the nondimensional parameters of the system through use of the Buckingham-\\pi. The approach is made more efficient by the use of centroids rather than comparisons to the full training data.", "questions": "1. Often, these surrogate methods you're describing are trained on simulation data which is already non-dimensionalized. How would you expect the performance of this method to be affected in this setting where new data is likely using the same characteristic scales? Given the is already dimensionally equivalent, will performance be affected? \n2. What's the justification for using the mean in place of characteristic scales? Often in fluids, these scales are relevant to the geometry of some feature in the system. Is this something that can be ignored in the current setting?\n3. Could you provide more detail on how the datasets were generated?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 3, "strengths": "1. It's a very sensible approach which tackles an important problem. \n2. The experiments are informative and show really strong performance. \n3. The authors' were clearly conscious of the impact of cost and found an approximation that works well. \n\nOverall, it seems the method is a natural and very promising approach for mitigating OOD performance, at least in the cases where the OOD is due to different dimensional choices could easily happen when applying a pretrained model on new data.", "weaknesses": "While there are some unanswered questions and aspects that could be tightened up which I'll list further down, the main reason for my current recommendation is that the presentation could use significant improvement, particularly on writing to a machine learning audience where a many readers will have very little experience with dimensional analysis. I think this can be easily rectified with some restructuring and building out examples better. Here are some concrete issues and suggestions:\n1. Many readers who work on neural surrogates are not going to be familiar with dimensional analysis (not arguing that this should be the case, but it is currently true), so when section 3 doesn't explain the basic concept well, subsequent sections will be harder to understand. If you expanded the worked example and using it through each stage of the section to describe what the fields are, what the units are, and then go through the current example showing how to extract $\\pi_{th}$ from them would make the statements significantly more concrete and easier to follow. \n2. It feels like concepts are often not explained in the place where they are presented. One would expect all of the methods to be explained in section 4 either mathematically or algorithmically, but section 4.4 just explains the uniform strategy as \"tunes the dominant scale while others fixed\". If this is supposed to be a method that doesn't require users to perform dimensional analysis on their own, how should they determine the dominant scale? How does this make the distribution uniform? \n3. Experiments are just equations right now. What types of common applications do these equations represent and why are they interesting test beds? How did you generate the data sets? How are initial conditions generated?\n\nOther issues:\n1. Experiments are currently fairly weak. These are two linear problems. It would be more interesting to see if the advantages from a linear projection method also holds for nonlinear problems. \n2. What are the test and train distribution of the invariants? For instance, it seems like q and k are shifted in the same pattern which wouldn't necessarily result in disjoint equations. It would be good to highlight where the method would be expected to fail as well. \n\nMinor:\n1. B not defined in Theorem 1. \n2. Random selection is mentioned, but not included in table 1.", "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": "M04VwoZxEN", "reviewer": "Reviewer_EMmu", "summary": "This paper proposes a Buckingham π-Invariant Test-Time Projection method to improve the out-of-distribution (OOD) robustness of PDE surrogate models such as FNOs, U-Nets, and CNNs. The key idea is that many OOD inputs differ only by unit or scale changes that should be physically equivalent under dimensional analysis. The authors therefore apply the Buckingham π theorem to define a log-space pi-group-preserving projection while aligning each test sample with its nearest training sample in parameter space. They proposed a minimization procedure for this alignment. Experiments on steady 2-D thermal conduction and linear elasticity show up to 90% reduction in OOD error with minimal computational overhead.", "questions": "- How sensitive is the method to the choice of π-group? Could the projection degrade performance if irrelevant or redundant groups are used? In many problems the pi groups are actually ratios between problem geometries, and it is unclear how they are to be chosen.\n\n- For heterogeneous domains, can local or hierarchical π values be used instead of global means?\n\n- How would the method behave on nonlinear or transient PDEs (e.g., Navier–Stokes, Burgers’, advection–diffusion)? Since pi scaling is used extensively in computational fluid mechanics, there should be examples from CFD for well-known pi groups \n\n- What is the computational cost for large training sets before centroid reduction, and how stable are results with different cluster seeds?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The paper adapts a century-old but fundamental physical principle (Buckingham π) into a practical, algorithmic test-time procedure for neural surrogates. \n- The combination of π-compliant projection and nearest-sample search is a creative way to transfer dimensional invariance into modern ML practice.\n- Training-free and model-agnostic: it can wrap around any pretrained surrogate without re-training or altering the loss, which makes it attractive for applied modeling.\n- Smooth minimization: the log-space formulation converts multiplicative scaling into a linear subspace problem, solved neatly by an orthogonal projection.\n- The idea may stimulate a broader discussion on how physical similarity and scale invariance can be enforced at inference rather than training time.", "weaknesses": "- Limited scope of experiments. Only simple, steady, linear PDEs (conduction and elasticity) are tested. It remains unclear whether the method holds for nonlinear, transient, or multi-physics systems.\n\n- Procedure feels overly elaborate for a scaling correction: The projection, clustering, and least-squares steps may appear heavy compared to straightforward normalization or nondimensionalization. There is a lack of exploration of when the procedure was worth, and when it is simply more favorable to cast more training data points.\n\n- Writing and exposition are often opaque: Key transitions between the physical reasoning, log-space math, and algorithmic steps are difficult to follow without prior familiarity.\n\n- No guarantee of a true physical neighbor: The nearest-sample search may project the test case toward an unrelated training sample if the π-space distribution is sparse or multimodal.\n\n- No discussion on mis-specified π-groups: The procedure assumes the chosen invariant is the correct one; the effect of using incomplete or incorrect π-groups is not analyzed.\n\n- Use of mean field values may fail for heterogeneous inputs: Collapsing distributed fields into global means ignores spatial structure, which can distort π values for systems with strong local variability.", "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)