Capacitor Chain Field Matching for Electrostatic Generative Modeling

Electrostatic Field Matching (EFM) constructs data-to-data generative transport by placing source and target distributions on oppositely charged plates in an augmented space. However, its single-capacitor geometry produces backward-oriented and exterior field lines, requiring stochastic branching for exact transport and complicating practical ODE-based inference. We propose Capacitor Chain Field Matching (C${}^2$FM), a framework that addresses these geometric limitations and revisits how electrostatic transport fields are learned. First, C${}^2$FM replaces a single pair of charged plates with an infinite periodic chain of alternating source and target distributions. The resulting symmetry confines field lines between adjacent plates. We prove that the ideal construction transports the source distribution to the target along forward-oriented field lines. Second, we derive a training objective with single-sample targets and theoretically grounded perturbation kernels, replacing EFM's multi-sample field estimates and heuristic perturbation scheme. Together, these contributions preserve EFM's electrostatic interpretation while enabling practical transport and training without multi-sample field estimation.