Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
Authors: Alexander Kolesov
S. I. Manukhov
Vladimir Vladimirovich Palyulin
Alexander Korotin
Venue: ICML 2025 poster (2025)
Keywords: generative models
distribution transfer
electrostatics
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: The New physics-inspired paradigm for Generative modeling

Abstract: We propose Electrostatic Field Matching (EFM), a novel method that is suitable for both generative modelling and distribution transfer tasks. Our approach is inspired by the physics of an electrical capacitor. We place source and target distributions on the capacitor plates and assign them positive and negative charges, respectively. We then learn the capacitor's electrostatic field using a neural network approximator. To map the distributions to each other, we start at one plate of the capacitor and move the samples along the learned electrostatic field lines until they reach the other plate. We theoretically justify that this approach provably yields the distribution transfer. In practice, we demonstrate the performance of our EFM in toy and image data experiments.

Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
Alexander Kolesov 12
Stepan I. Manukhov 13 Vladimir V. Palyulin' Alexander Korotin 12
Abstract
We propose Electrostatic Field Matching (EFM), a novel method that is suitable for both genera- tive modeling and distribution transfer tasks. Our approach is inspired by the physics of electri- cal capacitor. We place source and target distri- butions on the capacitor plates and assign them positive and negative charges, respectively. Then we learn the electrostatic field of the capacitor using a neural network approximator. To map the distributions to each other, we start one plate of the capacitor and move the samples along the learned electrostatic field lines until they reach the other plate. We theoretically justify that this approach provably yields the distribution transfer. In practice, we demonstrate the performance of our EFM in toy and image data experiments. Our code is available at https: /ttttbb.ttttbtt// com/ masttolleoo//iiellaatchh.i
2
1
E
X
P
x
)
x
x
3
3
3
2
2
2
2
Figure 1. Our Electrostatic field matching (EFM) method. Two data distrihutions P(x*) and Q(x ) x= RD are placed in the space R D+1 in the planes = 0 and = I,, respectively. The dis- tribution P(x*) assigned a positive charge, and the distribution Q(x - negative charge. These charges create an electric field ( where x = (x, =) RD+1 The lines the field begin at positive charges and end at negative charges. Movement along the electric field lines provably (see our Theorem 3.1) transforms the distribution P(x+) into the distribution ((x )
1. Introduction
The basic task of generative modeling is to learn transfor- mation between two distributions accessible by i.i.d. sam- ples. The typical scenarios considered are noise-to-data (Goodfellow et al., 2014) and data-to-data (Zhu et al., 2017). These are usually referred to as the unconditional data generation and data translation, respectively.
Diffusion Models (Sohl-Dickstein et al., 2015: Ho et al., 2020, DM) is popular class of generative models which is inspired by the nonequilibrium thermodynamics. The dif- fusion models consist of forward and backward stochastic processes (Song et al., 2021). The forward process cor- rupts the data via injection of Gaussian noise; the backward process reverses the forward process and recovers the data.
Physics is often at the heart of the principles of generative modeling. One of the first attempts to link generative mod- els and physics was made in Energy- Energy-Based models (LeCun & Huang, 2005, EBM). They parameterize data distribu- tions using the Gibbs-Boltzmann distribution density and generate data through simulation of Langevin dynamics (Du & Mordatch, 2019; Song & Kingma, 2021).
Poisson Flow Generative Models (Xu et al., 2022; 2023, PFGM) use ideas from the electrostatic theory for the data generation process, recovering an electric field between a hyperplane of the data and hemisphere of large radius.
'Equal contribution 'kooloovo Institute of Science and Tech- nology, Moscow, Russia Artiifcial Intelligence Research Institute, Moscow, Russia 'Lomonosov Moscow State University, Faculty of Physics, Moscow, Russia. Correspondence to: Alexander Kolesov <a..olesov @ skoltech.ru>.
Both DM and PFGM use physical principles to corrupt data, simplifying the data distribution to a tractable one. As result, they are only used directly for noise-to-data tasks.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 20255 by the author(s).
Recently, modifications of DM have appeared that can learn diffusion in data-to-data scenario. Diffusion Bridge Match-

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Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
ing (Shi et al., 2024; Albergo & Vanden-Eijnden, 2023; Peluchetti, 2023, BM) is SDE-based method that recov- ers the continuous- time Markovian process between data distributions. Flow matching (Lipman et al., 2023; Liu et al., 2023; Klein et al., 2024; Chen & Lipman, 2024; Xie et al., 2024, FM) is the limiting case of BM that learns ODE-based transformation between distributions.
can be written as:
1 (x x') E(x) = 4T |x - x'13 (xx)
(3)
The charge distribution g(x) can have values greater than zero (positive charge) or less than zero (negative charge).
However, there is no method based on electrostatic theory that can be applied to data-to-data translation tasks.
An electric field strength line is curve x(t) E RR, t e [a, bj C IR whose tangent to each point is parallel to the electric field at that point. In other words:
Contributions. We propose and theoretically justify new paradigm for generative modeling called Electrostatic Field Matching (EPM). It based on the electrostatic theory and suitable for both noise-to-data and data-to-data generative scenarios. We provide proof-of-con coofnoe--cncept experiments on low- and high-dimensional generative modeling tasks.
dx(tt E(x(t)), where t E [a. bj C R. dt
(4)
Electric field lines are the key concept in our work. The basic properties of these lines follow from Gauss's theorem and the circulation theorem formulated below. The rigorous formulations and proofs of the field line properties are given in Appendix A (for a more general D-dimentional case).
2. Background and Related Works
2.1. Basic physics
The electric field flux. Consider an element of area dS. is vector whose length is equal to the considered area dS and whose direction is orthogonal to this area. For closed surfaces, the direction is always selected outward. The electric field flux E through this element is d& = E dS, i.e., the inner product between vectors E and dS. The field flux through a finite surface 2 is given by
To understand the physics behind the electrostatic field matching method, let us recall some basic background from standard Maxwell's 3D-electrostatie and then generalize the case of D dimensions. Information on Maxwell's electrostatics can be found in any electricity textbook, for instance (Landau & Lifshitz, 1971, Chapter 5).
2.1.1. 1. MAXWELL'S S ELECTROSTATICS
f dΦ [f E.d..
(5)
The field of a point charge. Let a point charge g E IR be located at a point x' R². At a point x E IR3 it creates an
It indicates the density of field lines passing through Σ. Gauss's 's theorem (Landau & Lifshitz, 1971, $31). For any closed two-dimensional surface 0M, which bounds the set M C RR3 (see Fig. 2), the electric field flux is equal to the total charge enclosed by this surface:
electric field? E(x) E R equal to:
Exx) = q x x'
4u ||x
x 1
The superposition principle. If point charges q1, 42, .......... IN are located points x1, X2,..., Xy. they create independent fields E, (x), Eg()), .... Ev (x) at a given point e R All these charges together produce the following field:
E E.dS- ds = S.x q(x)dxx. àm
(6)
E(x)
Σ (x)
On 4T
(x
X,,
Intuitively, the Gauss's 's theorem states that the number of lines passing through any closed surface is determined only by the charge inside that surface (and is proportional to it).
x,,
In the general case, we are dealing with continuously distributed charge 9x). Then the superposition principle
In particular, it follows from Gauss's theorem that the elec- tric field line must begin at a positive charge (or at infinity) and end a negative charge (or at infinity). Lines cannot simply terminate in a space where there are charges, see Lemma A.3 and Corollary A.4 for extended discussions.
'All formulas are written in the Heaviside- units, where Planck's 's constant = 1, the speed of light e= 1. and the electric constant, which stands as a multiplier in Coulomb's 's law (see (1)), is k 1/4T.). TTii system of units is convenient for purposes because formulas look particularly simple in this system of units (the Gauss's theorem and the circulation theorem).
system of
Theorem on the electric field circulation (Landau & Lif- shitz, 1971, (26). For any closed loop E (Fig. 3) the electric field circulation is equal to zero:
'The meaning of electric field is as follows. If charge qu is placed in electric field, then the force acting go equals to F E. Using (1), obtain Coulomb's 's law of interaction of point charges: k uag (x- - x)), where = Tx
$ E dl = 0,
(7)
2

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Field Matching: Electrostatic Paradigm to Generate and Transfer Data
E(x)
9
(8)
Sp-1 x x'I
ds
where Sp-1 is the surface area of an (D 1) dimenssiona sphere with radius
E
The field of a distributed charge ((x) can be obtained by the principle of superposition as in (3) for 3D case:
Sax g(xx)xx
/ SD xx
x x
E(x) =
dxx.
(9)
The Gauss's theorem and the circulation theorem in D- dimensional space together ensure the following principal characteristics of electric field lines:
Figure An illustration of the Gauss's 's theorem.
(i) Electric field lines cannot terminate in points where there are no charges;
(ii) For system having zero total charge (f ((xxxxxx 0), electric field lines almost surely² start at positive charge and end at negative charge;
where dl is the length element of the closed loop f. This length element is a vector whose length is determined by the infinitesimal segment dl, see Fig. 3, whose direction is the tangent given point the curve L.
(iii) There are no electric field lines that form closed loops.
2.2. Poisson Flow Generative Model (PFGM)
The first attempt to couple electrostatic theory and genera- tive modeling is proposed by (Xu et al., 2022; 2023). The authors work with a D-dimensional data distribution.
E
They embed this distribution into (D 1)-dimensional space. The new point in this space can be written
dl
(I1, ..... DD. 2) (x, 2) x X D+1
(10)
l
The data is then placed on the hyperplane = 0 at R D+1
by applying x - X (x, 0). The data distribution is inter- preted then as a positive electrostatic charge distribution.
Figure 3. An illustration of the electric field circulation thcorem.
The intuition of the method that the charged points in the hyperplane which behaves at infinity as the field of a point charge. If point charge is placed inside a sphere Sxe with an infinite radius, then the flux density P Poc (-) through the surface of the sphere is distributed uniformly. For simplicity, the authors consider the hemisphere S (Fig. 4b, upper half of Sxo). The electric field lines define the correspondence between uniformly distributed charges on the surface of S and the data distribution Po( located in the hyperplane = = 0.
0 generate the electric field E(-)
follows from the circulation theorem that there are no field lines which form closed loops, see Lemma A.7.
2.1.2. D-DIMENSIONAL ELECTROSTATICS
The generalization of electrostatic equations for higher di- mensions appears discussions related to the influence of extra dimensions on physics (Ehrenfest, 1917; Gurevich & Mostepanenko, 1971; Caruso et al., 2023). The generaliza- tion modifies equations (6) and (7) by replacing R with RD and replacing dimensionality 2 of 0M with D - 1 in the Gauss's 's theorem. The definitions in (4) and the superpo- sition principle remain unchanged. The differences affect only the explicit expression for the electric field.
If massless point charge is placed in the electric field E(-) with field lines directed from (-) to I (-). then the charge moves along the lines to oc (.). This movement transforms the data samples from the complex distribution Po(.) into
"The term "almost surely" indicates that for randomly se- lected point from the first positively charged distribution, the prob- ability that its electric field line terminates the second negatively charged distribution (rather than at infinity) equals 1.
The electric field at the point x E R of a point charge q, which is located x' E RD equals to:
3

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Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
+2
E x
E
X ~ ((5+)
2 2= 0
Po (x
(a) Near the plate.
(b) Away from the plate.
Figure 4. PFGM concept. The original data have distribution (x(X), which is assigned positive charge that produces an electric field [()). Near the plate (Fig. 4a), the field lines can have a complex structure, while away from the plate (Fig. 4b) the charge looks like a point, and therefore the electric field uniformly distributed: x (((.))).
the simple distribution (-) on the hemisphere. The corre- sponding inverse transformation generates the data samples from uniformly distributed samples on the hemisphere. The inverse map movement along these field lines in the backward direction and is defined by the following ODE with electric field
charge density follows the distributions up to a sign. We then place these distributions on two D-ddmenniooall planes at distance L from each other (Fig. 1). This will produce an electric field with lines starting at one density and finish- key result-
ing at another. We prove Theorem 3.1 which shows that movement along the lines guarantees an almost sure transition from one distribution to another.
-E(-) being the velocity field:
dk(t) E(((1))-
(11)
dt
3.2. Formal theoretical justification
To recover the electric field E(-) in the extended (D + 1)- dimensional space, the authors propose to approximate it
Let P(xx) ) and Q(x ), with x D., be two data distribu- tions. We assign to the first distribution a positive charge
with neural network fo(-): : D+1 R D+1
q xx+) = Pxx*) and to the second distribution a negative charge 7 (x- = (x Note that the charge distri- butions are normalized such that f s (x+x(xx
First, they compute the ground truth electric field E(x) em- pirically at set of arbitrary (D + 1)-dimensional points x inside the hemisphere St through samples from Po(-) using (9). Second, the electric field is learned at X by mini- mizing the difference between the predicted fa (x) and the ground-truth E(x). Having learned the electric field E(-) in the (D 1)-dimensional space. they simulate ODE (11) with initial samples from Px. (.) until the spatial coordinate reaches O. Finally, they get samples Xy Po(-), where T is the end time of the ODE simulation.
1 and J q (x ddx 1, resulting in a total charge of zero.
We place these a and q in (D + dimiiennsional space. Each point in this space has the same form as in (10). More precisely, we place g² in the hyperplane
0, and in 2 == L (Fig. 1). One can think of it as a (D+1) -dimeesniin capacitor. The distributions can be written with Dirac delta function 8(.) as:
xx (x = (x)0(2), q = q (x, = (x)o(z L).
3. Electrostatic Field Matching (EFM)
(12)
This section introduces Electrostatic Field Matching (EFM), novel generative modeling paradigm applicable both noise-to-data and data-to-data generation grounded in elec- trostatic theory. In 3, we give an intuitive description of the method. In $3.2, we give the theoretical foundation of the method and the main theorem. In $3.3, we formulate the learning and inference algorithms of the EFM.
The electric field produced at the point x D+1 in the space plates consists of two summands:
E(x) = E. (x) Ex),.
(13)
where E (x) and E_ (x) the fields created by 9 (x and g (x respectively. The exact expression for these fields is given by (9) with replacement of D by D + 1:
3.1. Intuitive explanation of the method
Our idea to consider distributions as electric charge den- sities. One could assign positive charge values to the first distribution and negative charges to the second one, i.e., the
E. (x) =
Sp - $ #
(14)

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Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
Consider the field lines originating from the plate P(x+). Due to the total zero charge, these lines almost surely termi- nate on Qxx ), as established in Lemma A.6. The transport from P(.) to (.) can be initiated in two distinct directions: either toward (forward-oriented) or away from (backward- oriented) the target plate. While backward-orient exhibit greater curvature and longer paths, they still termi- nate in the target distribution, see Figs. 21a and 21b.
point Finally, we define the forward map as follows:
with probability "(xp) with probabiliyy 1 - v(x7)
Tx(xt)= = F
(16)
lines
The probability u(xp)) allows us to understand whether it needed to stop at the first intersection point Xp. or continue moving to the next point x'p (see Fig. 5). In particular, the probability value is proportional to the electric field flux.
We define the stochastic forward map TF from (Ppp(P) to supp(Q) (Q) through forward-oriented field lines.* For this, we consider point x = (x+, E + O slightly shifted
The stochastic backward map TB is constructed similarly using left limit e + O and backward-oriented field lines.
in the direction of the second plate. Let us move along the corresponding field line by integrating dx(() = dt
The complete transport is described by the random variable:
until coming to the second plate at a point X (xp: T.)-
Txxxxt) with probability p(x*), (T'(x") with probability 1 - m(x+);
Next, two situations may arise (Fig. 7):
T(x*)
(17)
1. The values of the field to the left and right of the plate 2 = L have different signs. Here the field is directed toward the plate from two different sides.
capturing both forward and backward trajectory endpoints for each x+ E supp((P) on the left plate, where:
2. The values of the field to the left and right of the plate have the same signs on the left and right, i.e., the field line crosses 2 = L and continues to the region > L.5
if E have same sign
m(xx) EE
(x
E have opp. signs
(18)
The value ,u(**) allows one to choose a forward or back- ward sets of lines with a probability proportional to the field flux in the corresponding direction (i.e., E or |E D. At the same time, is impossible to move backward, i.e., Eg > 0, the forward map TF is always chosen (x(x*) =
P
P
X F
F
Then, for this stochastic map, we prove the next theorem:
Theorem 3.1 (Electrostatic Field Matching). Let F(x*)
and Q(x ) be two data distributions that have compact support. Let x be distributed as P(x*). Then x = T(xt) is distributed as Qxx almost surely:
X
F
2 L
2-
Z
Z
If xxx ~ P(xx) = T(x+) =x- Q(x
(19)
E 2
E+
E
In other words, the movement along electrostatic field lines does indeed transfer P(x to Q(x ), as we intended. The proof of the theorem is given in Appendix B.
Figure An illustration of the forward map Tr.
In the first case, the movement ends. In the second case, it necessary to stop the movement at the point Xp with prob- ability v(xT) and continue the movement into the region with probability 1 =(xx), where
3.3. Learning and Inference Algorithm
To move between data distributions, it is sufficient to follow the electric field lines. The lines can be found from the trained neural network approximating the electric field E(x). In practice, we only use
1, E. (x))- (x,)
if E (x have opposite signs
if 1 (xF) have same sign
field lines, since backward-oriented ones have more curvature and require a larger training volume, see Appendices C.1 and C.3.
B. (NF)
(15) and (xp + se,), + 0 denote the left and right limits of the field value at the point X7 in z direction. The latter field line also reaches the target distribution Q, at
Training, To recover the electric field E(-) in (D + 1)- dimensional points between the hyperplanes, we approxi- mate it with a neural network fo(-) RRD D+1 The approximation requires setting of a training volume. We sample the value & from the uniform distribution 2/(0, I.)
'Here supp(-) denotes the support of distribution. 'The movement in the opposite direction is impossible since the projection F, of the field is positive in the interval = (0, T.).
5

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Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
and take two random samples xt and x Then, we get a new point via the following averaging and noising:
reaches = L again. However, in practice we simply stop the movement as accurate estimation of this probability is non-trivial, see Appendix C.2 for extended discussions.
x= + + -xt +
(20)
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, correspondingly.
where random E is obtained as follows. First, the noise & is sampled from ( o² (D++)) where is a hyperparameter. Then, the Euclidean norm ||e]] is multiplied by a normalized standard Gaussian vector m
Algorithm EFM Training
Input: Distributions accessible by samples:
P (6() and Q )²(2 - -L);
-= - [mll' m (0,
(21)
NN approximator fo(.) RR + RDDI
Output: The learned normalized electrostatic field so(.)
The noise term increases the training volume, thus, leading to the greater generalization ability of the neural network.
Repeat until converged: :
Sample batch of points x P(xx))(2);
We highlight that this interpolation is just one of the possible ways to define intermediate points, see Appendix C.4.
Sample batch of points x ~ Q(x )0(2 L);
Sample batch of times t ~~ u10, T)):
The ground-truth truth E(x) is estimated with (13). Specifically, the integral is approximated via Monte Carlo sampling of (9) by using samples from PP(x+) and (x"). Then we use a neural network approximation Sa (X) to learn the nor- malized ground-truth electric field ||[[)]] at points from the extended (D + 1)) dimmensiioaal space as, analogously to PFGM, we empirically found that this strategy works better than learning the unnormalized field. We learn f-(-) by minimizing the squared error between the normalized E(x) ground truth ||](|]|| and the predictions (x) with SGD, ie., the learning objective is
Sample a batch of noise ō with (21);
Calculate X = 1 + (1 1 )x + E;
Estimate F. (x) and E_ (x) through (9);
Calculate E(x) with (13);
B(x)
Compute C = 1Fx [e (x)
FG)
||2 + ming;
[E(]]]
Update by using 00
ac.
Algorithm 2 EFM Sampling
Input: sample x+ from P(x ((x)) step size AT > 0; the learned field fā(:) : RR D- + R P+1,
Output: mapped sample x approximating Q(x )8(2 L)
E [X]]
(x)
min. (22)
Set Xg- x for 7 E (0, AT, 2Ar, ...... L- Ar) do Calculate lă {x, = (x)(X=) )x))
Inference. Having learned the normalized vector field E(-)
Kr+Ar + (x,)2 )2 + (x,)) fØ ATIT + Ar]
BEO in the extended space with fa(-), we simulate the movement between hyperplanes to transfer data from Pxx
NL
to ((x ). For this, we run an ODE solver for (11).
4. Experimental Illustrations
One needs right stopping time for the ODE solver. We follow the idea of PFGM and exploit the following formula:
In this section, we demonstrate the proof-of-concept ex- periments with our proposed EFM method. We show a 2-dimensional illustrative experiment ($4.1), image-to- image translation experiment ($4.2) and image generation experiment (54.3) with the colored MNIST and CIFAR-10 datasets. Various additional aspects of EFM are studied in Appendices: the influence of the interplate distance L (App. C.1), training volume (App. C.4) and backward-oriented field lines (App. C.3), We give details of experiments in Appendix D.
dx dl
dz d(x, =)
dz, dz) = (Ez (x)E (x), 1)dz
dt dz
Eg
E X
1)dz Re (f) (x)x, 1)
[
(23)
where we denote [x(x) (ox()) and E(x) =
(E, E, (x)). In this new ODE (23), we replace the time variable with the physically meaningful variable 2. We start with samples from Pxxx), i.e., when = 0, and we stop when 2 reaches T. (= right plate) during the ODE path.
4.1. Gaussian to Swiss Roll Experiment
An intuitive first test to validate the method is to transfer between distributions whose densities can be visualized for comparison. We consider the 2-dimensional zero-centered Gaussian distribution with the identity covariance matrix as
According to definition (16) of the forward map Tp, when field line crosses the plane 2 = L, one should continue its movement with probability v(x;) until the line
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Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
x(x1)
TTx"1
T(x*)
(a) Samples from P(x which are placed on the left hyperplane = O.
(b) Samples from Q(x ), which (c) Mapped samplex by T(x*) for the (d) Mapped samples for T(x for the placed on the right hyperplane = L.
distance I. 6.
distance = 30.
Figure 6. Ilussrattive 2D Craussinn+Swiss Roll experiment: input and target distributions P(xt) and Q(x together with the result of the distribution transfer Icarned with our EFM method for distances L = 6 and L = 30 between the capacitor plates.
P(x+) and the Swiss Roll distribution as Q(x"), ). see their visualizations in Figs. 6a and 6b, respectively.
To show the effect of hyperparameter L in our EFM method, we do two experiments. In the first one, the samples from Qxx") are placed on the hyperplane = 6 (see Fig. 6c), while in the second one, we use L = 30 (see Fig. 6d). We show the learned trajectories of samples' movement along the electrostatic field in Figs. 8a and 8b, respectively.
When L is small, the electric field lines are more or less straight, see Fig. 8a. The learned normalized electric field fe(-) allows one to accurately perform the distribution trans- fer, see Fig. 6c. However, if the distance L between the hyperplanes large, the learned map recovers the target density Qxx") poorly (see Fig. 6d). Presumably, this oc- curs due to higher difficulty of the electrical field E(-) approximation for large interplane distance L, as well as the neccesity to consider the lines outside ie.. backward-oriented lines as well as the forward-oriented lines which go to the area 2 > (intersect the plate twice).
(a)
(b) L
Figure 8. Electric field line structure for the Gaussian Swiss Roll experiment with L = 6 and 30. can be seen that at large distances, the field lines are more curved than at small distances.
defined correspondence, see (7hu et al., 2017, Fig. 2).
We place colored digits on the left hyperplane = and colored digits 2 the right plate 2 10. We learn the normalized electric field between the plates and demonstrate input- trans-transl pairs in Fig. 10a and 10b, respectively. Also, we show how the translation happens in Fig. 7.
(0, T.),
4.2. Image-to-Image Translation Experiment
For comparison, we add the results of the translation of the popular ODE-based Flow Matching (FM) method (Liu et al., 2023; Lipman et al., 2023; Tong et al., 2023) in Fig. 10c. The key difference between our method and FM is that FM matches fime-conditional transformation (velocity), whereas method matches to space-conditional trans- formation (electric field). Interestingly, FM does not always
Here we consider the image-to-image translation task for transforming colored digits 3 to colored digits 2 (Gushchin et al., 2024, $5.3). The data is based on the conventional MNIST images dataset but the digits are randomly colored. We consider unpaired translation task, i.c., there is no pre-
3
3
2
2
2
2
2
2
2
2
3
3
2
2
2
2
2
2
2
2
2
3
3
3
3
2
2
2
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2
2
2
Figure The sampling trajectories of our EFM method in image- to-image translation experiment, see $4.2.
Figure 9. The sampling trajectories our EFM method in noise- to-image generation experiment, see $4.3.
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Field Matching: Electrostatic Paradigm to Generate and Transfer Data
3
3
3
3
3
3
2 2 2
2
2
2
2
2
2
2
3
3
3
3
3
3
2
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8
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2
(a) Samples from P(x which placed the left (b) Samples from our approximation of Q(x ), located plate 0.
(c) Samples from FM's approximation of Q(x )
on the right plate 10.
located on the right plate = = 10.
Figure 10. amage-to-lmage translation. Pictures from the initial distribution, the result of applying our EFM method as well as the Flow
Matching method presented respectively.
3 2 8
8 2
3
8
8
7
3
55903
8
6 0
5
9
0
1
8 4 3 6 0
5
6
0
6
3
0
(a) White noise samples from F(x+), which are placed (b) Samples from our approximation of Q(x located on the right plate = 30.
(c) Samples from PFGM's approximation of Q(x
on the left plate = 0.
sirullated from hemisphere with the learned field.
Figure 11. Noise-to- coise-to-Image generation. Pictures from the initial distribution (Fig. 1a)), the result of our EFM method (Fig. 11b) as well as the PFGM method (Fig. 11c) are presented.
(a) White noise samples from P(x+) which are placed on (b) Samples from our approximation of Q(x located (C) Samples from PFGM's approximation of Q(x sim- the left plate U.
the right plate 500.
ulated from bemisphere with the leamned field.
Figure 12. Noise-to-lmage generation. Pictures from the initial distribution (Fig. 12a), the result of EFM method (Fig. 12b) well the PFGM method (Fig. 12c) are presented.
accurately translate the shape and color of the initial digits 3. In Appendix D, we demonstrate the results of the translation obtained with several other relevant methods.
For completeness and comparison, show the results of generation of (Xu et al., 2022, PFGM) which is also based on the electrostatic theory, see Fig. Ic and Fig. 12c.
4.3. Image Generation Experiments
5. Limitations
We also consider the task of generating 32 x 32 full colored digits from the MNIST dataset. For this task, we place white noise on the left hyperplane 2 = 0 and colored digits on the right plate z = 30. After learning the electric field between the plates, we demonstrate mapping to
MNIST.
Our EFM method has several limitations which open promis- ing avenues for future work and improvements.
Influence of dimensionality. In high dimensions, our algo- rithm may need to operate with small values. Specifically, the multiplier 1/l|x - x the electric field formula (9) can produce values comparable to machine precision as increases. As result, the training of our method may become less stable. Note that the same holds for PFGM.
the target distribution Q(x") in Fig. 1lb. We qualitatively that using only forward lines allows to recovers the target distribution Q(x)) of colored digits well. The sampling trajectories for our EFM are shown in Fig. 9.
CIFAR-10. The experiment with qualitative comparisons on more complex CIFAR-10 data is presented Fig. 12b. We place the white noise on the left plate z 0, while images placed on the right plate 500.
The impact of interplate distance L on the field. The larger the interplate distance L is, the more curved and dis- perse the electric field lines become, see, c.g., Fig. 8b. With an increase of the distance the electric field has to be ac- curately learned in a larger volume between the plates.
8

---

Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
careful selection of the hyperparameter L may be impor- tant when applying our method. Further discussion of the choice of parameter L can be found in Appendix C.I.
geometries. In The Twelfth International Conference on Learning Representations, 2024.
De Bortoli, V., Korshunova, I., Mnih, A., and Doucet, Schrodinger bridge flow for unpaired data translation. Advances in Neural Information Processing Systems, 37: 003384-10344)) 2024.
Defining the optimal training volume. Our training ap- proach involves sampling points x and x from the dis- tributions, interpolating them with (20) and noising them with (21). This allows us consider an intermediate point between the plates (20) learn the electrostatic field. There may exist more clever schemes to choose such points, see Appendix C.4. It is a promising question of further research.
Du, Y. and Mordatch, I. Implicit generation and modeling with energy based models. Advances in Neural Informa- tion Processing Systems, 32, 2019.
The problem of lines going beyond E (0, L). Modeling the movement along the electric field lines, e.g., forward stochastic map Tp (16), theoretically requires considering the lines which cross the boundary practice, we ignore this requirement and stop when L. which may lead to incorrect learning of the target data distribution. We discuss this in Appendix C.2.
Ehrenfest, P.
In what way does it become manifest in the fun- damental laws of physics that space has three dimensions? Proceedings of the Amsterdam Academy of Sciences, 20: 200-209, 1917.
L. However, in
Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., and Bengio, Y. Generative adversarial networks. Communications of the ACM, 63(1 I): 139--144 2014.
Backward-oriented field lines. Each plate emits two dis- tinct field line sets: forward-orienten and muckward-oriente trajectories. Our practical implementation exclusively uti- lizes forward-oriented lines due to their reduced training volume requirements. Nevertheless, backward-oriented tra- jectories remain theoretically significant, particularly for en- suring complete support coverage of Q(x"), see App. C.3.
Gurevich, L. and Mostepanenko, V. On the existence of atoms in n-dimensional space. Physics Letters A, 35(3): 201-2022 1971.
Gushchin, N., Kolesov, A., Korotin, A., Vetrov, D. P., and Burnaev, E. Entropic neural optimal transport via diffu- sion processes. Advances Neural Information Process- ing Systems, 36, 2024.
Impact Statement
This paper presents work whose goal advance the field of Machine Learning. There are many potential societal consequences of our work, none of which feel must be specifically highlighted here.
Ho, J., Jain, A., and Abbeel, Denoising diffusion proba- bilistic models. Advances in neural information process- ing systems, 33:6840-6857 2020.
Acknowledgement
Kingma, D. and Ba, J. L. Adam: A method for stochastic gradient descent. In ICLR: international conference learning reprexentations, pp. 1-15, ICLR US., 2015.
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 000000C3133255PFF0002 and the agreement with Skoltech N 10-20200-202--33.
Klein, L.., Krämer, A., and Noé, F. Equivariant flow match- ing. Advances in Neural Information Processing Systems, 36, 2024.
References
Landau, L. D. and Lifshitz, E. M. The Classical Theory of Fields (Volume of A Course of Theoretical Physics). Pergamon Press, 1971.
Albergo, M. and Vanden-Fijnden, E. Building normaliz- ing flows with stochastic interpolants. In ICLR 2023 Conference, 2023.
LeCun, Y. and Huang, F. J. Loss functions for discriminative training of energy-based models. In International work- shop on artificial intelligence and statistics, pp. 206-213. PMLR, 2005.
Caruso, F., Oguri, V., and Silveira, F. Still learning about space dimensionality: From the description of hydrogen atom by generalized wave equation for dimensions / Z 3. American Journal of Physics, 91(2)1153-158, 2023.
Lipman, Y., Chen, R. T., Ben-Hamu, H., Nickel, M., and Le, M. Flow matching for generative modeling. In The Eleventh International Conference on Learning Repre- sentations, 2023.
Chen, R. T. and Lipman, Y. Flow matching on general

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Field Matching: Electrostatic Paradigm to Generate and Transfer Data
Liu, X., Gong, C.,, and qiang liu. Flow straight and fast: Learning generate and transfer data with rectified flow. In The Eleventh International Conference Learning Representations. 2023.
Peluchetti, S. Diffusion bridge mixture transports, schrÄyjinneer bridge problems and generative model- ing. Journal of Machine Learning Research, 1 51, 2023. URL htttp://jmls org/ papers/ /22/ 23 0527 html.
Shi, Y., De Bortoli, V., Campbell, A., and Doucet, A. Diffu- sion schrödinger bridge matching. Advances in Neural Information Processing Systems, 36, 2024.
Sohl-Dickstein, Weiss, E., Maheswaranathan, N., and Ganguli, S. Deep unsupervised learning using nonequi- librium thermodynamics. In International conference machine learning, pp. 2256-2265. PMLR, 2015.
Song, Y. and Kingma, P. P. How to train your energy-based models. arXiv preprint arXiv:2101.03266 2021.
Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Er- mon, S., and Poole, B. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021.
Su, X., Song, J., Meng, C., and Ermon, S. Dual diffusion implicit bridges for mmgeeto--image translation. In The Eleventh International Conference on Learning Repre- sentations, 2023.
Tong, A., Malkin, N., Huguet, G., Zhang, Y., Rector-Brooks, J., Fatras, K., Wolf, G., and Bengio, Y. Conditional flow matching: Simulation-free dynamic optimal transport. arXiv preprini 23X.:23322.0482.. 2(3), 2023.
Xie, T., Zhu, Y., Yu, L., Yang, T., Cheng, Z., Zhang, S., Zhang, X., and Zhang, C. Reflected flow matching. arXiv preprint ar Xiv:22405. 16577, 2024.
Xu, Y., Liu, 7.., Tegmark, M., and Jaakkola, T. Poisson flow generative models. In Proceedings of the 36th Inter- national Conference on Neural Information Processing Systems, pp. 16782-16795, 2022.
Xu, Y., Liu, Z., Tian, Y., Tong, S., Tegmark, M., and Jaakkola, T. Pfgm++: Unlocking the potential of physics- inspired generative models. International Conference on Machine Learning, pp. 38566-38591, PMLR, 2023.
Zhu, J..Y., Park, T., Isola, P., and Efros, A. A. Unpaired image-to-ine tommeee--mmaeeee translation using cycle-consistent adver- sarial networks. In Proceedings of the IEEE international conference on computer vision, pp. 2223-2232, 2017.
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Field Matching: Electrostatic Paradigm to Generate and Transfer Data
A. Properties of D-dimensional electric field lines
In this auxiliary section, we formulate and prove the basic properties of electric field lines in D-dimensions.
Definition A.1 (The flux of the electric field). The flux of the electric field E through an area dS is called d$ = d... The flux through a finite surface Σ is defined as an integral:
+
d
[[
E. ds.
(24)
Definition A.2 (The stream surface, Fig. 13). Consider closed piecewise smooth curve I' placed in an electric field. Consider field line passes through each point of this contour. The set of these lines is called stream surface (tube).
$ lat
T
r
E
Figure 13. The electric field flux through an arbitrary stream tube.
Lemma A.3 (Conservation of the field flux.). The electric field flux is conserved along a stream surface if there are charges inside that surface.
Proof. Consider arbitrary stream tube (Fig. 13). Let cut this tube by two arbitrary piecewise smooth cross sections. As result, we obtain a closed surface. Let us denote the flow through the side cross sections by 4 and 42, respectively. Note that the normal for closed surfaces is directed outwards. Near the right end of the tube, the normal and the electric field are co-directional, and near the left boundary, they have opposite directions. Therefore, $ = - - We have to prove that 0 = 22. The full flux is a sum of fluxes through the ends of the tube and through its lateral surface:
T Full
+1 + 42 + D
(25)
By the definition of a stream tube, the flux through the lateral surface must be zero: D = 0, because the field E and the normal to the lateral surface are orthogonal. Thus, + ffult = Φ + t2 = D2 - Ty. From Gauss's 's theorem (6), we derive:
[ʃ E- E.ES / ((x)xxx =0. O. 8M
$full
(26)
It follows that d/- = 2-
Corollary A.4 (Impossibility of line termination in empty space). An electric field line cannot terminate in empty space.
Proof. Otherwise, the field flux inside the current tube surrounding the termination point will not be conserved. Intuitively, near given point the field will enter, but will not exit, which is impossible in region with charge density.
Lemma A.5 (The electric field flux from a point charge.). Let us assume that a surface Σ is seen at solid angle Ω from point charge yo (Fig. 14). The electric field flux through this surface is equal to:
'By definition, the solid angle Q at which the given surface Σ is visible from a given point Q is cqual to the area of the projection of this
surface onto the unit sphere centered at this point. Mathematically, the solid angle element R² is defined as dl
r-48
-
where r E R is vector drawn from Q to considered point on surface E, r = ||||] dS1 = d5 cos a dS F.

---

Field Matching: Electrostatic Paradigm to Generate and Transfer Data
Σ
dS
E
C
n
Ω
Figure 14. The field flux of a point charge øo through an arbitrary surface E seen at solid angle f.
9052
Φ
(27)
SD
Proof. Divide Σ into small surface elements dS. The total flux is the integral over the entire surface, + = -sx dd. By
definition of flux (Definition A.1), and according (8) we have:
ds
god5
dΦ E dS EdS cos EdS_
(28)
SD-1 PD1 Sp-1"
where a is the angle between E and dS, dS = dS cos a. Then after integration over the solid angle, we obtain (27).
Lemma A.6 (Electric field lines start and end points). Let an electrically neutral system, J q(xx)xx 0, be bounded in space, i.e. q(x) has compact support. Then the electric field lines must begin at positive charges and end at negative charges, except perhaps for the number of lines of zero flow.
Proof. Let denote the diameter of the system by == max x ( (||x x'I)). Consider our system at large distance I, >>> f. Let the small parameter = E/T. < characterize this distance. We use the multipole decomposition of an electric field (Landau & Lifshitz, 1971, 540-41):
r = (()) + +E() +... = Epoint + (1ا5
E LD-2
(29)
Multipole decomposition a representation of an electric field as a sum of contributions from point sources of different multipoles, monopole (0) Epsint: dipole E(1), quadrupole E(2) etc. allows describing the field large distances from the system, simplifying the calculations by neglecting the contributions of higher multipoles. In problem, the multipole decomposition shows that the field at a large distance from the system looks like field of a point charge located at the origin. All other contributions can be neglected. And therefore, in the limiting case + 0, L + oo the formula (27) can be used. Since there no limit on the increase of L, one can achieve an approximation accuracy as high as one needs.
Then + E dS QΩ/Sp-1 0, because Q = (xx)xx = the total charge. Hence, flux through any surface located distance |r|| from the system is zero:
/ E ds when ||||| 2 L + oc.
(30)
Suppose non-measure- zero set of field lines escapes to infinity. These lines would generate flux & in contradiction = 0. Thus, all field lines originate positive charges and terminate negative ones, except for set of measure L
12

---

Field Matching: Electrostatic Paradigm to Generate and Transfer Data
E
Figure 15. Closed loop an electric field line. This situation impossible due to circulation theorem.
Lemma A.7 (Absence of closed electric field loops). Electric field lines cannot form closed loops (as shown in Fig. 15).
Proof. Assume, to the contrary, that closed loop exists. Then E 2 0 holds everywhere along & (since E and dll are co-directed), and the equality zero of dl cannot be satisfied everywhere (otherwise is not closed loop and even a field line). Consequently, the circulation integral satisfies:
j E dl >
However, the circulation theorem (7) requires:
$ E dl
yielding contradiction.
B. Proof of Electrostatic Field Matching theorem
In this section, we prove several auxiliary lemmas, and then move on the main theorem. Let IP(-), Q(-) be two D-dimensional data distributions having a compact support, located the planes 0 and = L. in RD+1 respectively, Lemma B.1 (On the relation between distribution and field values). For any point in the support of distribution P(-):
E (x)
E
(
(31)
Proof. Consider an infinitesimal volume dS the support of P. Consider closed surface, cylinder with infinitesimal indentation in different directions in the plane = 0, Fig. 16.
Pds
d$
E dS
d
E
dS
d lat
Figure 16. The considered area.
The flux through this surface consists of three summands: d the flux in the positive direction of the z-axis; d the flux in the negative direction of the z-axis; and d the flux through the lateral surface:
d kull
d + d + d in F ds
F. dS
(32)
13

---

Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
Here iat since the height of the cylinder under consideration can be made small as wants (infinitesimal of higher order than dS). In turn, d④ Fy dS has negative sign due to the fact that the normal to the closed surface is directed outward, i.e in the opposite direction from the axis Then, due to the Gauss's ' theorem (6):
d = (E E )dS = Pds
(333
which proves the lemma.
Let (x, 1==1 be a sample of points distributed over P. Let dn be the number of points from the sample that fall in the volume dS in the support P. Then dn = dn + dng, where dnp is the number points dS that correspond to the mapping Tp (that is, the movement along forward oriented lines; see (17)), and dn corresponds TB-
Lemma B.2 (First lemma on the flow). Let the field near the element dS have different signs on different sides: E > 0
and Eg < 0. Then:
dnf
0.s.
E dS
d4
Tt
n=+00
(34)
dn
a.s.
|Ez dS
dd
11 71+000
where n-
denotes the almost sure convergence, and dl F,B denotes the electric field flux near the plate surface the forward and backward directions.
Proof. According to the multiplication rule probability and the law large numbers:
dnr a.s.
(probability of choosing Tp) (probability of falling in dS)
π
(35)
E (E + 1)dss = -E dS = ddr
1 P(X)SS= E. + 1
In the second equality, the definitions of probability u(-), see (18), and Lemma B.1 are used. The case dn is proved similarly.
Lemma B.3 (Second lemma on the flow). Let P, Q, He dS, dn have the same meaning as in Lemma B.2. Let E and Er have the same sign near dS (i.e., E > 0). Then
dn
d after
d befores
(36)
7-
where before is the field flux through the current tube supported on dS immediately before crossing the plane dS e supp P.),, and d after
affer is the flux after crossing.
Proof. By the probability product theorem, the strong law large numbers, Lemma B.1, and the definition of flux:
dn a.s. 1-000
(x = 1 PP (Et EE )dS draaner d&Pbefore-
(37)
Remark. This statement implies that when the field crosses the plane I containing a charge (proportional to dn/n), the field flux must increase by dn/n.
Lemmas B.2 and B.3 address the behavior near the distribution F. Similar statements are valid for the behavior near Q When moving along field lines, we eventually reach the plane = T.. At this point x two different scenarios may occur:
1. and Fg (x-) have opposite signs. Then the field line motion terminates in this case.
14

---

Field Matching: Electrostatic Paradigm to Generate and Transfer Data
2. (x-) and E (x have the same sign. Then some number dn' S dn of lines inside current tube (corresponding sample {x] i=1
P)) must terminate, while the others should continue moving.
This portion dn' can be found from the line termination property in Q
Lemma B.4 (Line Termination). If E (x-) and E (x-) have the same sign upon crossing = L, the number of lines terminating on z satisfies:
dn'
0.5 after + d befores
dD
(38)
n
Remark. When the field crosses the plane Q containing a charge (proportional to dn'/n) /n), the field flux must decrease by dn'/n.
Proof. Consider the current tube before it intersects the plane 2 = L. Let us denote the number of lines inside dribefore As result the intersection 2 = L, some of the lines dn' stop moving, while some of the lines d'aatter continue moving. view of the first Lemma B.2 on flow, as well as the conservation of flow inside the current tube (Lemma A.3):
drbbefore
3.5.
d'Pbefore.
(39)
n
Then, by virtue of the law of large numbers and the fact that drivetore = dn' + dMaatters we have:
dn'
dribefore
(probability of termination).
u(x d'Prefore
11
220
72000
(40)
E
E+
E. "S' = -E5- - -E+ )dS' -dd adPare + d
Fiz
We now proceed to prove the main theorem. Theorem B.5 (Electrostatic Field Matching). Let (xxx and Q(x L be distributed as F. Then x = =T(x*) is distributed as Q(x almost surely:
) be two data distributions that have compact support.
xx*~ = T(x+) = ~ Qxx
(41)
Proof. 1. Let {x, Vi be a sample of points distributed according to PP-). Applying the mapping T()), we obtain the corresponding points on the target distribution: {x, 1 = (T(x'* )
2. Consider a D-dimensional area element d5' on Q(-). Let dn' denote the number of points x; within this region. We define the empirical distribution:
dn' n
2n (x yds
(42)
Our goal is to prove:
(.)+) almost surely Q(-).
(43)
3. The dn' points arrive at Q(-) through two pathways (Fig. 17). forward- (dn's) and backward-oriented (dn'g) trajectories:
dn' = dn'p + dn'g.
(44)
4. Consider the current tube terminating at d5' corresponding to forward-oriented arrivals (dr(p). Since P(-) and Q(-) have compact supports and the total charge related to the system the two plates is zero, by Lemma A.6, these tubes must start from P..)) almost surely.
15

---

Field Matching: Electrostatic Paradigm to Generate and Transfer Data
==L
IdS"
dn'r
dn B
dn
x
Figure 17. An element of volume dS' is selected on the distribution Q(-). dn' is the number of points x;" falling into this volume. Some points came to the distribution ((-) from front (denote them by drlr)), and some came from back (dn'))
5. During the motion against field lines from Q to P, multiple crossings of & = 0 and/or 2 = may occur (N = 0, or 2, see Fig. 18). Denote the intersection points:
x
X0
XN
XN+1 = x
(45)
P
P
Q
P
Q
P
x
X
X1
X1
X
X2
X1
x
X
X
N =
=
N
N = 2
Figure 18. Intersection points.
Their corresponding area elements are
ds' = dS. + dS1 + .... dSN + dSN+1 = dS.
(46)
Point counts in these areas
dn' dno + dni + + dnn - dn dn,
(47)
where dny (k: = 0, ..., + 1) is number points from sample {x or from map (T(x)]m= inside the volume dS, near point X5, that corresponds considered motion inside current tube.
6. The dnp are not arbitrary but related by flux conservation. Only the charged planes (z 0 or L) can alter the count:
At 2か 0: Line count increases by drg-
At zą Ta Line count decreases by dny.
16

---

Field Matching: Electrostatic Paradigm to Generate and Transfer Data
It can be written mathematically as:
N+1 =0,
Σ A=
(-1)
(48)
where
0 if 2 0,
fk
(49)
1
if Zk L.
7. Due to the first Lemma B.2 on flow:
dn
dn
uss.
de N+ d
(50)
4-000
8. Due to the second Lemma B.3 on the flow, and because of the line termination Lemma B.4:
dn
(-1)/+
a.s.
dlt after, k d& before, d.
(51)
n2 n-000
9. According to the law of conservation of flux along the tube (Lemma A.3) from to X;-1i
d4 after dd debetorrekee be
(52)
10. Whence obtain chain of equalities:
N+1
dn'p
+... + (-1) du1 + (-1)6)
dnp dn
fa du
+
n
n
n2
n
k=
dn'p
(-1) dn +... + (-1)5N dnn
+ dn
(53)
n
n
n
+00
as.
dd a 142before.1 + ... + dll a d'Ptefore... N + d2 N+1
n+
1 + +0+.. ... + =d
Consequently,
dn's
d
(54)
Note that this property is not obvious in general, since the points dn' are obtained by applying the mapping T(+) to certain sample of points, while the flux d'' determined by the true value the field, which is defined by the entire distribution of charges TP and Q
Similarly, it can be proven that
dn'p
d
(55)
n
2=000
11. Then, by virtue of the Gauss's 's theorem (6), we finally have
dn'
dn'p
dn'g
d4'
d4's
dS
B
ass
B
QdS.
(56)
This completes the proof.
17

---

Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
C. Extended discussion of limitations
C.1. The problem of choosing the hyperparameter L
The interplate distance between the left and right plates the capacitor is the key tuning hyperparameter that influences the performance. In the $4.1, we provide an experiment on Swiss roll dataset that demonstrates the impact of L on the performance (Fig. 8). With neural-network--bssee approaches, the electric field becomes less recoverable as the hyperparameter increases as, informally speaking, the required training volume also grows.
We also carry out another example to show the influence of the hyperparameter L in noise-to-image generation CIFAR-10 experiment (Fig. 19). It can be seen that increasing L up to the value of 5000 leads to significant decrease in the generation quality. At the same time, no significant differences are observed for the average values of L, = 50, 500. The theoretical ideal L that minimizes the approximation error remains unknown and represents an important direction for future work. Hovewer, usually we choose L of an order comparable to the standard deviation of the data distributions.
(a) 50
(b) = 500
(c) 5000
Figure 19. The influence of the interplate distance I on the performance of the Noise-to-Imann Image generation of CIFAR-IO dataset. Pictures EFM method with 50 (Fig. 19a), L = 500 (Fig. 19b) as well as L = 5000 (Fig. 19c) are presented.
C.2. The problem of lines going beyond 2 L
When moving from P to Q along the field lines, we eventually reach the plane L.. Two different scenarios may occur:
1. (L) and ET (L) have opposite signs. The field line motion terminates in this case.
2. (L) and E (L) have the same sign. Then, the movement should end at this point with probability v(x"), and with probability -(x"), ), the movement should continue (recall (15)).
Overall, one needs stochastic choice for terminating and/or continuing the movement. To make this choice, it is necessary to calculate the value of the field E (L) to the left and right of the plate L at the point of interest and calculate v(x
Suppose that the situation of continuing movement is realized. This motion should be performed by integrating dx(tt E(x E( until the line on the target distribution again. Note that such motion should indeed end on the distribution Q (Lemma A.6), and also that further movement is impossible (since F (L) < 0, and between plates (0, L) we always have Eg in particular E__ (L) 0, i.c., the first situation is realized). An example of considered problem is shown in Fig. 20. Note that in practice, the field is calculated from sample of data (rather than from continuous distribution), it is difficult to calculate E accurately. Therefore, in practice we always stop the movement at the first intersection.
C.3. Backward-oriented field lines
Transport between plates oceur via two distinct trajectory classes (Fig. 21a):
Forward-oriented trajectories (red lines): Initial motion directed toward the target plate
Barkeeedd trajectories (black lines): Initial motion receding from the target plate
Note that due to the Lemma A.6, both series must end at the second distribution. The difference between the two series is the greater curvature of the backwerd-oriented series compared the forward-oriented one. From the practical point of view,
18

---

Field Matching: Electrostatic Paradigm to Generate and Transfer Data
-5
-000)
-10
Txx
Figure 20. A toy experiment Gaussian 2Gaussians. The two Gaussians are significantly separated from each other, so a large number of lines flying out of the region I. is observed. Further integration along the field lines still leads to the target distribution.
using backward-oriented series is less useful than forward-oriented, This is not only due to the curvature of the lines, but also due to the significantly larger training volume (Vbacwaard shaded in black in the figure is much larger than red Vioward)-
We also note another feature related to the training volume. In Fig. 21a, we can see that lines starting the periphery of the data distribution are particularly curved, and may partially leave the training volume (points A, B, D, H))
E
F
backward
forward
Q(x
P(x+)
Q(x
Pxx+))
(a) Two series of field lines: Barward-directed (red lines), and bookward-directed (blue lines). Blue lines require much larger training volume Visttaard than red lines, which take up Vieward The figure also shows the field lines starting from the peripheral distribution points that can leave the training volume (points A, B, C, D. F,G. H).
(b) Two series of field lines: forward-directed (red lines), and bockward-directed (blue lines). can seen that forward- oriented (red) lines starting at the distribution P(-) may not completely cover the target distribution (?)).
Figure 21. An illustration of forward-oriented and Beckward-orienterr lines.
When source (P) and target (Q) distributions exhibit significant mean shifts (pp # (Q), the number forward-orieen lines extending beyond the J. boundary can be quite large. In such situation, there appears a region on the second distribution
19

---

Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
that is not covered by forward-oriented lines (in Fig. 21b red lines do completely cover the second distribution).
C.4. Training Volume Selection
Our electrostatic framework allows flexible training volume definitions between plates. The (20) formula is not the only possible training volume option (different from the flow matching, which uses this particular interpolant for loss construction). In Fig. 22, comparison of the two approaches is shown:
Linear interpolation between samples of IP and Q using (20)
Uniform cube mesh between plates.
Figure 22 showcases the equivalent performance for both approaches in Gaussian-to-S -oSiiss-rol transport.
y-4
y-9
0I2SASO
(a) Training volume for our EFM method that is defined by interpolation from (20).
(h) Training volume for our EFM method that is defined by uniform cube mesh between plates.
VpYO
0123A56
0123450
(c) The learned trajectories of our EFM with training volume defined in 20.
(d) The learned trajectories of our EFM with training volume detined by the cube mesh.
Figure 22. Learned trajectories of our EFM method with different training volumes on Gaussian-to-Swiss roll experiment.
D. Experimental details
We aggregate the hyper-parameters of our Algorithm 1 for different experiments in the Table 1. We base our code for the experiments on PFGM's code /tttps://giteee
Toy experiments. In the 2D illustrative example 1$4.1), we make the inference by constructing the ODE Euler solver for the equation 11 with the iterative scheme (see Alg.2)
20

---

Field Matching: Electrostatic Paradigm to Generate and Transfer Data
Image data experiments. In the case of the Image experiments ($4.2 and $4.3), we the RK45 ODE solver provided by https: /ocs..c.....//oo/oocooo//..//... http:///////////p/////////eeeeee
scci scipy- integrate. RK45. html for the inference process with the hyper-parameters rtol=le-4, atol=le-4 and number of steps (NFE) equals to 100. Also, we use Exponential Moving Averaging (EMA) technique with the ema rate decay equals to 0.99. As for the optimization procedure, we use Adam optimizer (Kingma & Ba, 2015) with the learning rate = 2e and weight decay equals to le-4.
Evaluation of the training time for our solver on the image's experiments (see $4.2 and $4.3)takes less than 10 hours on single GPU GTX 1080ti (11 GB VRAM).
Experiment
Batch Size
L
NFE,Num Steps
A,LR
Weight Decay
Gaussian Swiss-roll
2
1024
20
2e-3
0.001
Colored MNIST Translation (3++2)
3072
64
10
100
2e-4
le-4
0.01
Colored MNIST Generation
3072
64
30
100
2e-4
le-4
0.01
CIFAR-10 Generation
3072
500
100
2e-4
le-4
0.05
Table Hyper parameters of Algorithm different experiments, where 13 is the dimensionality of task, I. is the distance betwenn plates and it used for the definition between plates (see $3.3).
Baselines. We use the source code https:///ithut: coom/Wewweeerr/ffgggg for running PFGM in our ex- periments. We found the following values of hyper parameters are appropriate for us: 7 5, 0.3, le - 3, (Xu al., 2022) for details. Also, we utilize the source code of Flow Matching (FM) from the github page https://githin htt.ghhhb...o//teteeeeeeeeeeeeee in $4.2. We also add extra baselines: Cycle-GAN from the github page bttps://githhe
/eatteeeeemiiiinneeeeeeieeeeaiia for experiments
com/ junyanz/ com/ http///uuaann/n/ea ddib and DSBM from attps://gi com/yuyange shi /ddbmmmtomm.r.... Their results are shown in Figure 23.
pytorch Caal--yc-llleaaaeeeee and Baee---nn--iiie -ix2pixx DDIB from https:
2 2
2
2
2.2
2
2
2
2
122
22
2
2
2
2 2
2
2
2
2
2
2
2
2
2
6
2
2 2
2 2
2
2
2
2
2
2 2
2
2
2
(a) Samples Cycle-GAN approximation of ((x")
Samples approximation of Qx")
Samples approximation Q(x")
Figure 23. Image-to-Image translation experiment (Colored MNIST datasct, 3 + 2). The results of alternative translation methods:
Cycle-GAN (Zhu et 2017), DDIB (Su et al., 2023) and DSBM (De Bortoli al., 2024)
21


Reviews:
[{"review_id": "Y9WiUBoyk1", "reviewer": "Reviewer_cYJh", "summary": "In this work, the authors propose Electrostatic Field Matching (EFM), which transforms between two distributions in the same space by placing the two distributions on two parallel plates with opposite charge, training a neural network to predict the electric field in the space between the two plates, and trace a test charge through the field from one plate to the other. Beside theoretical justifications, some interesting results on random generation and translation between different digits on a colored MNIST dataset are shown as a proof of concept.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer, thank you for reviewing our paper. Below we answer your questions.\n\n**(Q1) [...] Experiments on more complex datasets[..]some quantitative evaluations would also help.**\n \nFollowing your request, we include **additional experiments** with more complex data such as CIFAR-10 and the requested full color-MNIST dataset. For qualitative analysis, we demonstrate our EFM's results as well as PFGM's performance. We ask you to get acquainted with Fig. 1 and Fig. 3 that are available via the anonymous link  https://drive.google.com/file/d/1DTbQR_GNah7hVGGnDF822aD96iWxjR-k/view?usp=sharing. For quantitative analysis, we calculate FID/CMMD scores and compare with another methods. Please, familiarize yourself with the the [answer to YMMk](https://openreview.net/forum?id=9dHilxylvC&noteId=1boFlYDdKN).\n \n**(Q2) [...] accurate approximation of the  field using Monte Carlo integration? [...] In high dimensions, field contributions from distant charges decay rapidly with distance [...]**\n\nIf we correctly understand your question, your ask about the accurate approximation of $E(\\widetilde{x})$ in Eq. (11) for a given $\\widetilde{x}$ via Monte Carlo sampling. Since Monte Carlo integration provides an unbiased estimation of an integral and its variance does not depend on the dimensionality, this estimation might be used for accurate approximation of the field. However, the more samples are used in batch for estimation the lower variance. We conduct **additional experiments** with different batch sizes and its influence on our EFM's performance. Please, familiarize yourself with the Fig. 2 via the main link.\n\n**(Q3) [...] dipole effect [...] Some field lines initially travel backward [...] reach the target plate from behind. [...] how can we ensure accurate tracing of outward field lines?**\n\nIndeed, each plate emits two distinct sets of field lines. One set is directed towards the second plate and another oriented in the opposite direction. Crucially, the properties of electric field lines (LemmaA.7) ensure that both sets almost surely terminate on the opposing plate. The primary distinction lies in their geometric trajectories: the forward-directed lines exhibit smaller curvature and reach the target plate more efficiently (faster) than their backward-oriented counterparts. From a practical point of view, prioritizing forward-directed trajectories is advantageous for computational efficiency while remaining theoretically sound due to Theorem~3.1, which guarantees almost sure transport from $\\mathbb{P}(\\cdot)$ to $\\mathbb{Q}(\\cdot)$.\n\nTo address field lines extending beyond the plate boundary ($z > L$) before reaching $\\mathbb{Q}(\\widetilde{\\mathbf{x}}^-)$, one may use the following natural criterion to distinguish the valid termination on $\\mathbb{Q}(\\cdot)$ from the transient crossings of $z = L$:\n\n\\begin{equation}\n    \\begin{cases}\n        E_z(z \\to L^-) = E_z(z \\to L^+) & \\implies \\text{The line goes away past the distribution}, \\\\\n        E_z(z \\to L^-) \\neq E_z(z \\to L^+) & \\implies \\text{Valid termination on } \\mathbb{Q}(\\widetilde{\\mathbf{x}}^-).\n    \\end{cases}\n\\end{equation}\n\nThis criterion derives from:\n 1) Field continuity along current tubes (Lemma A.3, Corollary A.4),\n 2) Boundary field behavior (LemmaA.5): Near a plate, the field is determined by the charge of this plate and is directed away from the plate in the case of positive charge (and towards the plate in the case of negative charge, see lemma A.5). Therefore the direction and magnitude of the field must change in the case if we arrive at the target distribution.\n \nThus, discontinuities in $E_z$ explicitly signal successful transitions to $\\mathbb{Q}(\\cdot)$, while continuity indicates a need for further integration.\n\n**(Q4) matter of parameter L [...]**\n\nWe conduct **additional experiment** that deals with the influence $L$ on the performance (see Fig.4 via the link). The more distance between plates the worse approximation of the field. If L is too small, field is recoverable , but there is \"edge effect\" and it might has influence on performance.\n\n**(Q5) PFGM [...] the underlying theory is the same [...]**\n\nWe respectfully disagree. Our significant theoretical advancement compared to PFGM is using the following fundamental property of electric field lines in $D$-dimensional space. The field lines starting from the positive charge distribution $\\mathbb{P}(\\cdot)$ almost surely terminate in the negative charge distribution $\\mathbb{Q}(\\cdot)$ (Lemma A.7). This property — *previously not considered in electrostatic generative models* — combined with field flux conservation along current tubes, formally establishes (our Theorem 3.1) that field line trajectories transport samples between $\\mathbb{P}$ and $\\mathbb{Q}$. This constitutes our main theoretical contribution.  \n\n**(Q6) Is it safe to ignore field lines that start out facing the wrong direction?**\n\nYes, it is safe due to our main theorem. See the answer to the **Q3** for details.", "claims_and_evidence": "While the theoretical proofs for the principles appear sound, the experimental results are limited, as acknowledged by the authors who present them as toy examples without making broad claims.", "methods_and_evaluation_criteria": "I have several concerns about the method. First, how can we ensure accurate approximation of the ground truth field using Monte Carlo integration? In high dimensions, field contributions from distant charges decay rapidly with distance, making nearby charges significantly more important. However, in high dimensions, Monte Carlo samples are unlikely to land near these crucial regions, potentially leading to poor sampling of the most important areas. Though I acknowledge PFGM faces a similar challenge.\n\nSecond, parallel plates with opposite charges create a dipole effect, where field lines inevitably spread into the surrounding space rather than staying confined between the plates. Some field lines initially travel backward, away from the target plate, make extensive detours, and eventually reach the target plate from behind. While PFGM can focus on one side due to symmetry, the lack of symmetry here raises questions about whether we can safely consider only the \"right\" side while ignoring the \"wrong\" side. Even if we can focus on one side, field lines can travel far from the plates before returning—a significant issue in high dimensions where most directions are approximately perpendicular. When training samples are created by interpolating between two random points on the plates, they remain between the plates. This raises the question: how can we ensure accurate tracing of outward field lines? PFGM avoids this issue since it intentionally allows field lines to extend to infinity.\n\nFinally, there's the matter of parameter L. The Swiss Roll experiment demonstrates L's crucial role—larger values cause more field lines to deviate sideways, negatively affecting results. While smaller L values likely have their own drawbacks, the authors should provide guidelines for determining appropriate L values.", "theoretical_claims": "I didn’t find any specific issues.", "experimental_designs_or_analyses": "Experiments on more complex datasets are desirable. It seems to me that the jump from “placing the target distribution on a changed plate surrounded by an infinite sphere and run a test charge” (PFGM) to “placing two distributions on two parallel plates with opposite charges and run a test charge” (proposed method) is not a big one, especially considering that the underlying theory is the same, so I would say this concept is in large parts already proven. Thus, I think mere proof-of-concept experiments are not sufficient. Even if we are only getting simple examples, there should at least be the result of random generation on the full color-MNIST dataset, instead of just a subset featuring the same digit.\n\nIn addition, some quantitative evaluations would also help.", "relation_to_broader_scientific_literature": "Not evaluated", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "No", "other_comments_or_suggestions": "The authors dedicate one and a half pages to basic physics concepts that, given their elementary nature indicated by the section heading, could be condensed. This space would be better used for additional experimental results.", "questions_for_authors": "1.What is the effect if L is too small?\n2. Is it safe to ignore field lines that start out facing the wrong direction?", "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": "QNXcqr8V6v", "reviewer": "Reviewer_d6Dp", "summary": "The paper proposes Electrostatic Field Matching (EFM), a method for generative modeling and distribution transfer based on electrostatic principles. EFM generalizes the Poisson Flow Generative Model (PFGM) by enabling mapping between arbitrary distributions. It represents source and target distributions as charged capacitor plates and learns the electrostatic field between them using a neural network.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the valuable comments. Please find the answers to your questions below.\n\n**(Q1) [...] quantitative evaluations or comparisons [...] alternative approaches from the literature (e.g., SB-based, GAN-based). CIFAR [...]**\n\nFollowing your request, we include **additional experiments** with more complex data such as CIFAR-10. For qualitative analysis, we demonstrate our EFM's results as well as PFGM's performance. See Fig.3 via the anonymous link https://drive.google.com/file/d/1DTbQR_GNah7hVGGnDF822aD96iWxjR-k/view?usp=sharing. For quantitative evaluation,  we calculate FID and CMMD[1] on test of CIFAR-10 and compare with PFGM, DDPM, DDIM and GLOW. Please see the table with results in the [answer to YMMk](https://drive.google.com/file/d/1DTbQR_GNah7hVGGnDF822aD96iWxjR-k).We see that in image generation our performance is comparable to PFGM. *At the same time, we remind that our method is also capable of performing data-to-data transfer while PFGM is not capable of doing that*.\n\nIn accordance of your request of comparison with GAN and SB-based methods, we also demonstrate the performance of our approach with SB-based (DDIB,$\\alpha$-DSBM ) approaches as well as GAN-based (Cycle-GAN ) for unpaired data setups on Colored MNIST. Please, familiarize yourself with Fig.6 via the afore mentioned link. We also see that our method demonstrates the highest FID and CMMD among compared approaches.\n\n| Metrics/Method | EFM (our) | FM    | DSBM | DDIB | GAN  |\n|----------------|-----------|-------|------|------|------|\n| FID            | **4.45**  | 19.87 | 7.21 | 8.24 | 4.57 |\n| CMMD           | **2.37**  | -     | 4.02 | 4.11 | 2.45 |\n\n**(Q2) The manuscript primarily follows the derivations from [1].**\n\nWe respectfully disagree. Our significant theoretical advancement compared to PFGM is using the following fundamental property of electric field lines in $D$-dimensional space: field lines starting from the positive charge distribution $\\mathbb{P}(\\cdot)$ almost surely terminate in the negative charge distribution $\\mathbb{Q}(\\cdot)$ (Lemma A.7). This property — *previously not considered in electrostatic generative models* — combined with field flux conservation along current tubes, formally establishes (our Theorem 3.1) that field line trajectories transport samples between $\\mathbb{P}$ and $\\mathbb{Q}$. This constitutes our main theoretical contribution enabling data-to-data transfer.\n\n**(Q3) In line 250, the authors mention that the inference process is summarized in Algorithm 1, but the algorithm only describes the training procedure.**\n\nThanks for pointing that out. The inference algorithm is just a simulation of the learned ODE as described in Section 3. For convenience of the reader, we will add a separate algorithm box with inference procedure in the final version of the paper.\n\n**(Q4) I wonder if, in Algorithm 1, t should be sampled from Uniform(0, 1) to ensure proper interpolation between x+ and x-**\n\nThe uniform sampling $t \\sim \\text{Uniform}(0,1)$ represents no more than a particular training volume selection strategy for interpolating between $\\widetilde{\\mathbf{x}}^+$ and $\\widetilde{\\mathbf{x}}^- $, controlling the density of training points along the interplanar axis. As demonstrated in Fig. 5 (link from Q1) for the Gauss-to-Swiss-roll experiment, both the proposed uniform training volume (Algorithm~1) and conventional cubic lattice initialization produce nearly identical electric field line configurations.\n\n**(Q5) How does the batch size influence the stability of the training? [...] batch size of 1?**\n\nTheoretically, it is possible to train our method with a batch size equal to 1. However, Monte Carlo integration has a higher variance for a small number of samples. Following the question, we conducted an extra experiment with different batch sizes. We demonstrate that performance almost stops to increase with batch sizes exceeding 64, see Fig. 2 in extra material (link from Q1)\n\n**(Q6) I wonder if samples that are on the periphery of the P distribution would not be pushed away and land far away from Q distribution as a result?**\n\n This scenario is precluded by Theorem 3.1, which guarantees transport between distributions $\\mathbb{P}(\\cdot)$ and $\\mathbb{Q}(\\cdot)$. While peripheral samples exhibit greater curvature compared to central trajectories due to boundary effects, this geometric difference does not prevent their convergence to $\\mathbb{Q}(\\cdot)$. The continuity of field lines (Lemma A.7) ensures termination on the target distribution almost surely.  \n\n**(Q7) [...] source code [...] unable to run. [...] functions not included in the provided package**\n\nWe will fix this in the final code version.\n\n**Concluding remarks.** We would be grateful if you could let us know if the explanations we gave are satisfactory. If so, we kindly ask that you consider increasing your rating. We are also open to discussing any other questions you may have.", "claims_and_evidence": "The authors introduce a new generative approach and present proof-of-concept experiments. Their claims are supported by visual evaluations.", "methods_and_evaluation_criteria": "The benchmarks are relevant but very limited. Only toy datasets (Swiss Roll and MNIST) are used, with no quantitative evaluations or comparisons to other methods from the literature.", "theoretical_claims": "I did not find any issues with the theoretical claims in the paper. The manuscript primarily follows the derivations from [1].\n\n\n[1] Xu, Yilun, et al. \"Poisson flow generative models.\" Advances in Neural Information Processing Systems 35 (2022): 16782-16795.", "experimental_designs_or_analyses": "Lack of reproducibility: The code in the Appendix imports functions that are not included in the provided package.\n\nThe paper is presented as a proof-of-concept without quantitative experiments, providing only visual results of the method. The details of the visual experiments are given in Appendix C and source code.", "relation_to_broader_scientific_literature": "The proposed method extends the idea introduced in [1] by replacing the source distribution—from a known uniform distribution projected onto a (D+1)-dimensional hemisphere—with any arbitrary distribution placed on a hyperplane parallel to the target distribution.\n\nThis concept of constructing the vector field from one distribution to another is known as a Schrodinger Bridge in Diffusion Models nomenclature. \n\n[1] Xu, Yilun, et al. \"Poisson flow generative models.\" Advances in Neural Information Processing Systems 35 (2022): 16782-16795.", "essential_references_not_discussed": "The references in this paper are relevant.", "other_strengths_and_weaknesses": "Strengths:\n- The authors propose a creative method that uses Poisson Flow [1] as a Schrödinger Bridge for interpolating between two arbitrary distributions.\n- The authors claim that the method works with unpaired datasets.\n- The paper is well-written and easy to follow, with the Related Works section being particularly well-structured.\n\n\nWeaknesses:\n- There are no quantitative evaluations of the method; the authors provide only visual results.\n- The method is not compared to alternative approaches from the literature (e.g., SB-based: [2, 3, 4], GAN-based: [5]).\n- The authors use only toy datasets (Swiss Roll, MNIST). Using more complex datasets (like CIFAR or ImageNet) would help determine whether the estimated field is sensitive to the dimensionality of the data.\n- In line 250, the authors mention that the inference process is summarized in Algorithm 1, but the algorithm only describes the training procedure.\n- The source code utilizes functions not included in the provided package, making it impossible to reproduce the method.\n\nWeaknesses 1-3 limit the work to a proof-of-concept, reducing its overall contribution.\n\n[1] Xu, Yilun, et al. \"Poisson flow generative models.\" Advances in Neural Information Processing Systems 35 (2022): 16782-16795.\n[2] Su, Xuan, et al. \"Dual diffusion implicit bridges for image-to-image translation.\" arXiv preprint arXiv:2203.08382 (2022).\n[3] Kim, Beomsu, et al. \"Unpaired image-to-image translation via neural schr\\\" odinger bridge.\" arXiv preprint arXiv:2305.15086 (2023).\n[4] De Bortoli, Valentin, et al. \"Schrodinger Bridge Flow for Unpaired Data Translation.\" Advances in Neural Information Processing Systems 37 (2024): 103384-103441.\n[5] Zhu, Jun-Yan, et al. \"Unpaired image-to-image translation using cycle-consistent adversarial networks.\" Proceedings of the IEEE international conference on computer vision. 2017.", "other_comments_or_suggestions": "I wonder if, in Algorithm 1, t should be sampled from Uniform(0, 1) to ensure proper interpolation between x+ and x-.", "questions_for_authors": "- As far as I understand, the field is estimated based on all samples from the batch. How does the batch size influence the stability of the training? Is it possible to train such a model with the batch size of 1?\n- Unlike PFGM, your approach does not project the Q distribution on the (D+1)-dimensional hemisphere, but places it on z=L hyperplane. I wonder if samples that are on the periphery of the P distribution would not be pushed away and land far away from Q distribution as a result? \n- The hyperparameter L seems crucial, but it is difficult to tune. Based on the experiments, if the data has higher dimensionality, L should be increased. Is it possible to propose a function that assigns an L value for any given number of dimensions D?", "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": "jrkKT8AQ3W", "reviewer": "Reviewer_aVB4", "summary": "The paper introduces Electrostatic Field Matching (EFM), a novel generative modeling framework inspired by the physics of an electrical capacitor. In EFM, source and target data distributions are assigned positive and negative charges on two parallel plates, and a neural network is used to learn the resulting electrostatic field. By moving samples along the field lines from one plate to the other, the method provably transforms the source distribution into the target distribution. This approach is versatile, addressing both noise-to-data and data-to-data generation tasks, and its theoretical guarantees and experimental results on toy and image datasets position it as a compelling alternative to existing diffusion and flow-based models.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer, thank you for reviewing our paper. Below we answer to your questions and comments.\n\n**(Q1) [...] performance on high-dimensional or complex tasks (real-images) [...] testing on more diverse and\ncomplex datasets [...] quantitative results are lacking..**\n\nIt is worth noting that our main goal is to demonstrate proof-of-concept of our method in the experimental section. We agree that providing additional generating experiments with more complex datasets enhances the understanding of method's performance. Further scalability of the method is a promising avenue for future research. \n\nNevertheless following your request, we include **additional experiments** with more complex data such as CIFAR-10. For qualitative analysis, we demonstrate our EFM's results as well as PFGM's performance. We would like to ask you to get acquainted with Fig. 3 that is available via the anonymous link https://drive.google.com/file/d/1DTbQR_GNah7hVGGnDF822aD96iWxjR-k/view?usp=sharing.\n\nFor quantitative analysis, we calculate FID and CMMD scores on test part of the aforementioned datasets and compare our method with PFGM, DDPM, DDIM and GLOW. Firstly, we demonstrate quantitative performance of our method on full Colored MNIST dataset. We see that our method outperforms other approaches on full Colored MNIST, reaching the lowest FID and CMMD.\n\n| Metrics/Method | EFM (our) | PFGM | DDPM | DDIM | GLOW |\n|----------------|----------|------|------|------|------|\n| FID            | **0.92** | 1.88 | 2.18 | 2.23 | 25.9 |\n| CMMD           | **1.47** | 2.28 | 2.68 | 2.85 | -    |\n\nSecondly, we demonstrate quantitative performance of our method on CIFAR-10 dataset. We see that in image generation on CIFAR-10, our performance is comparable to PFGM. *At the same time, we remind that our method is also capable of performing data-to-data transfer while PFGM is not capable of doing that*.\n\n| Metrics/Method | EFM (our) | PFGM       | DDPM | DDIM | GLOW  |\n|----------------|-----------|------------|------|------|-------|\n| FID            | 2.62      | **2.48**   | 3.17 | 4.16 | 48.9  |\n| CMMD           | **1.87**  | 1.93       | 2.98 | 3.25 | -     |\n\nAdditionally, we quantitatively and qualitatively compare the performance of our approach with SB-based (DDIB ,$\\alpha$-DSBM ) approaches as well as GAN-based (Cycle-GAN ) for unpaired data setups on Colored MNIST. Please, familiarize yourself with Fig.6 via the afore mentioned link. For quantitative analysis, we demonstrate FID and CMMD metrics on Colored MNIST dataset for our and compared methods.\n\n| Metrics/Method | EFM (our) | FM    | DSBM | DDIB | GAN  |\n|----------------|-----------|-------|------|------|------|\n| FID            | **4.45**  | 19.87 | 7.21 | 8.24 | 4.57 |\n| CMMD           | **2.37**  | -     | 4.02 | 4.11 | 2.45 |\n\nWe see that our method demonstrates the highest FID and CMMD among compared approaches.\n\n**(Q2) [...] sensitivity to hyper-parameters  [...] The inter-plate distance** \n\nWe conduct additional experiment that deals with the influence $L$ on the performance ( see Fig.4 via the link). The more distance between plates the worse approximation of the field. If $L$ is too small, field is recoverable , but there is \"edge effect\" and it might has influence on performance.\n\n\n**Concluding remarks.** We would be grateful if you could let us know if the explanations we gave are satisfactory. If so, we kindly ask that you consider increasing your rating. We are also open to discussing any other questions you may have.", "claims_and_evidence": "The method’s performance on high-dimensional or complex tasks (real-images) is not convincingly demonstrated, and its sensitivity to hyper-parameters—such as the inter-plate distance—raises concerns about practical robustness.", "methods_and_evaluation_criteria": "This paper primarily generates visual results, while quantitative results are lacking.", "theoretical_claims": "Yes, I have checked definitions and theorems in 3.1.", "experimental_designs_or_analyses": "The method is evaluated on three tasks:\n1. Gaussian-to-Swiss Roll Experiment: We consider a 2-dimensional, zero-centered Gaussian distribution with an identity covariance matrix as P(x+), and a Swiss Roll distribution as Q(x−).\n2. Image-to-Image Translation Experiment: This task involves transforming colored digit 3 into colored digit 2.\n3. Image Generation Task: This involves generating 32×32 colored images of digit 2 from the MNIST dataset.", "relation_to_broader_scientific_literature": "Shaul, Neta, et al. \"Flow Matching with General Discrete Paths: A Kinetic-Optimal Perspective.\" arXiv preprint arXiv:2412.03487 (2024).", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "1. The methods are well-motivated by electrostatic theory and the chosen evaluation tasks—using toy datasets and colored MNIST experiments—are standard for proof-of-concept generative modeling. \n\n1. However, further testing on more diverse and complex datasets could strengthen the practical validation.", "other_comments_or_suggestions": "I would like to raise rating if fair quantitive results and the evidence that this method can be scaled can be presented.", "questions_for_authors": "None", "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": "VBFB8Ak0aL", "reviewer": "Reviewer_YMMk", "summary": "The paper proposed a mechanism to train generative models. The main idea is to regard each data sample as a charge. The training process involves training networks to predict fields (gradients of potentials). The sampling process is done by solving an ordinary differential equation (moving data samples along fields).", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear reviewer, thank you for  reviewing our paper. Below we answer your  questions and comments.\n\n**(Q1) Only some visual comparisons with PFGM are provided. However, a comparison with RF/FM is necessary. Only toy datasets like 2D point sets and MNIST are provided.**\n\nIt is worth noting that our main goal is to demonstrate proof-of-concept of our method in the experimental section. We agree that providing additional generating experiments with more complex datasets enhances the understanding of the method's performance. Further scalability of the method is a promising avenue for future research. \n\nNevertheless following your request, we include $\\textbf{additional experiments}$ with more complex data such as CIFAR-10. For qualitative analysis, we demonstrate our EFM's results as well as PFGM's. We would like to ask you to get acquainted with Fig. 3 that is available via the anonymous link https://drive.google.com/file/d/1DTbQR_GNah7hVGGnDF822aD96iWxjR-k/view?usp=sharing.\n\nFor quantitative analysis, we calculate FID/CMMD scores on test part of the aforementioned datasets and compare our method with PFGM, DDPM, DDIM, GLOW. Firstly, we demonstrate quantitative performance of our method on full Colored MNIST dataset. We see that our method outperforms other approaches on full Colored MNIST, reaching the lowest FID/CMMD.\n\n| Metrics/Method | EFM (our) | PFGM | DDPM | DDIM | GLOW |\n|----------------|----------|------|------|------|------|\n| FID            | **0.92** | 1.88 | 2.18 | 2.23 | 25.9 |\n| CMMD           | **1.47** | 2.28 | 2.68 | 2.85 | -    |\n\nSecondly, we demonstrate quantitative performance of our method on CIFAR-10 dataset. We see that in image generation on CIFAR-10, our performance is comparable to PFGM. *At the same time, we remind that our method is also capable of performing data-to-data transfer while PFGM is not capable of doing that*.\n\n| Metrics/Method | EFM (our) | PFGM       | DDPM | DDIM | GLOW  |\n|----------------|-----------|------------|------|------|-------|\n| FID            | 2.62      | **2.48**   | 3.17 | 4.16 | 48.9  |\n| CMMD           | **1.87**  | 1.93       | 2.98 | 3.25 | -     |\n\nIn accordance of your request of comparison with  FM, we quantitatively and qualitatively demonstrate the performance of our approach with SB-based(DDIB, $\\alpha$-DSBM ) approaches as well as GAN-based (Cycle-GAN ) for unpaired data setups on Colored MNIST. Please, familiarize yourself with Fig. 6 via the afore mentioned link. For quantitative analysis, we demonstrate FID and CMMD metrics on Colored MNIST dataset for our and compared methods.\n\n| Metrics/Method | EFM (our) | FM    | DSBM | DDIB | GAN  |\n|----------------|-----------|-------|------|------|------|\n| FID            | **4.45**  | 19.87 | 7.21 | 8.24 | 4.57 |\n| CMMD           | **2.37**  | -     | 4.02 | 4.11 | 2.45 |\n\nWe see that our method demonstrates the highest FID and CMMD among compared approaches.\n\n**(Q2)  [...] rectified flow and flow matching (RF/FM) [...] the training loss is just a scaled version of RF/FM.**\n\nWe think that there might be a misunderstanding of our EFM method. In fact, we do not have a lot in common with flow matching (FM), except the fact that we learn an ODE to generate or transfer data. In particular, all the theoretical derivations and motivation totally differ.\n\n- We work in $D+1$ dimensional space and learn a static (non time-dependent) vector field to transfer data, while the field in FM is time-dependent and is in $D$-dimensional space.\n\n- The FM defines the interpolation $tx + (1-t)y$ between data samples $x$ and $y$ from two different data sets to regress a velocity field on $y-x$, where $t$ is sampled from a standard uniform distribution. In fact, the usage of this particular interpolant is principle for their loss construction. In turn, in our case, this interpolation is just a $\\textit{technical}$ way to define some inter-plate points $\\widetilde{x}$ between data distributions where to approximate Coulomb field $E(\\widetilde{x})$ at these points. In principle, we can use any other way to define the intermediate points.\n\nTo illustrate the aforementioned fact, we conduct the following 2-dimensional experiment with Swiss roll dataset. In the first case, we define the inter-plate points via the interpolation $tx + (1-t)y$ and approximate Coulomb field there. In the second case, we define uniform cube mesh between plates. The result of experiments demonstrates that the performance does not depend on interpolation. Please, familiarize yourself with the Fig.5 via the main link.\n\nSince the geometry of data distributions is sufficiently difficult in high-dimensional spaces, we use Eq. (19) as the way to define intermediate points just to cover a bigger volume of space.\n\n**Concluding remarks.** We would be grateful if you could let us know if the explanations we gave are satisfactory. If so, we kindly ask that you consider increasing your rating. We are also open to discussing any other questions you may have.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Only toy datasets like 2D point sets and MNIST are provided.", "theoretical_claims": "The idea sounds interesting at first sight. However, when I review the training algorithm, they looked very similar tthe o existing diffusion model design. \n\n1. The interpolation in Eq 19 is exactly the one used in rectified flow and flow matching (RF/FM), except for the random noise term. \n2. Eq 11 and Eq 16 are used for the target truth in training the network. The term in Eq 11 involves a term (x-x') which is also the one used in RF/FM, except for the normalizing term. Thus the training loss is just a scaled version of RF/FM.", "experimental_designs_or_analyses": "Considering the similarity between the paper and flow matching, I would like to know about some ablation studies.\n\n1. Is the term in Eq 19 important? In flow matching's training, we do not need the term. I still do not know if it is necessary to make the design work.\n2. The results are weak. Only some visual comparisons with PFGM are provided. However, a comparison with RF/FM is necessary.", "relation_to_broader_scientific_literature": "The method is inspired by fields in physics. However, the method is just a complicated version of rectified flow. It is hard for me to judge the quality of the method based on the current draft.", "essential_references_not_discussed": "Yes", "other_strengths_and_weaknesses": "No", "other_comments_or_suggestions": "After reading the paper, I have no interest in using the method in my projects. The designs are similar to rectified flows but more complicated. If the authors can ablate the designs, I would be more interested in the paper.", "questions_for_authors": "No", "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:

Meta-review: This paper presents an interesting variation of modeling the electrostatic interaction between two distributions by placing them on parallel plates with opposite charges. The conceptual idea is creative and well-motivated, drawing inspiration from physical analogies in a meaningful way.

One reviewer raised concerns regarding the novelty of the approach, noting similarities with rectified flows. However, the authors provided a satisfactory explanation during the rebuttal phase, clarifying the distinctions and contributing to a better understanding of their approach.

The primary weakness lies in the empirical evaluation. The initial results were limited to relatively simple benchmarks, which made it difficult to fully assess the method's effectiveness. The authors addressed this by providing more comprehensive results during the rebuttal stage. While these additions improved the evaluation, the method has not yet been demonstrated on high-dimensional datasets, which limits the scope of its practical applicability.

In conclusion, while the idea is novel and the theoretical framing is compelling, the empirical validation remains somewhat limited. Therefore, I am marginally in favor of accepting the paper, acknowledging its potential and the constructive efforts made during the rebuttal.
Decision: Accept (poster)