Interaction Field Matching: Overcoming Limitations of Electrostatic Models

Abstract

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

Figures (18)

• Figure 1: Electrostatic Field Matching kolesov2025field and our Interaction Field Matching (IFM) concepts. Two $D$-dimensional distributions $\mathbb{P}(\cdot)$, $\mathbb{Q}(\cdot)$ are placed in $\mathbb{R}^{D+1}$ at $z=0$ and $z=L$(a) In EFM, the distributions are interpreted as charges creating a capacitor-like electric field. Movement along these field lines transfers the distributions, but requires consideration of all directions of the field lines. (b) Our IFM is a generalization of the EFM to arbitrary interactions between charges. One possible realization of IFM is motivated by the strong interaction between quarks. This realization does not have backward-oriented lines and has a smaller curvature of lines.
• Figure 2: Limitations of the EFM & comparison with IFM. (a) The toy experiment ($1\rightarrow 2$ Gaussians) shows that even some forward-oriented field lines can leave $z>L$. These trajectories have increased length and curvature. Moreover, the transfer along only the forward-oriented lines does not cover the target distribution (green point cloud does not coincide with the red one). (b) Our realization of IFM (\ref{['SFM_realization']}) does not have the above mentioned problems: the field lines between the planes are almost straight, they do not extend beyond $z\!>\!L$ and are enough to cover the entire target distribution.
• Figure 3: Comparison of electrostatic interaction between charges $q^\pm$ (left) and strong interaction between quarks $q, \bar{q}$ (right). At small distances, the strong interaction resembles the electromagnetic interaction, but as quarks separate, the field lines straighten into a string.
• Figure 4: An illustration of the flux conservation.
• Figure 5: Illustration of forward, backward lines.

Theorems & Definitions (19)

• Lemma 3.1: On the field lines
• Example 3.2: The electrostatic field
• Theorem 3.3: Interaction Field Matching
• Theorem 3.4: Properties of our interaction field
• Definition A.1
• Lemma A.1: Generalized Gauss theorem
• proof
• Corollary A.2
• Lemma A.3: On field lines
• proof