ICML 2025

Field Matching: an Electrostatic Paradigm to Generate and Transfer Data
Alexander Kolesov * 1 2 Stepan I. Manukhov * 1 3 Vladimir V. Palyulin 1 Alexander Korotin 1 2
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 an 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 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. Our
code is available at https://github.com/
justkolesov/FieldMatching
1. Introduction
The basic task of generative modeling is to learn a 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.
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-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).
*Equal contribution 1Skolkovo Institute of Science and Tech-
nology, Moscow, Russia 2Artificial Intelligence Research Institute,
Moscow, Russia 3Lomonosov Moscow State University, Faculty of
Physics, Moscow, Russia. Corr...