Just Add More Capacitors: Eliminating Flux Leakage in Electrostatic Field Matching

Electrostatic field-line methods provide a physics-inspired view of generative modeling, originating in noise-to-data generation with Poisson Flow Generative Models. Electrostatic Field Matching (EFM) extends this idea to data-to-data generative modeling by placing source and target distributions on oppositely charged plates in an augmented space. However, the single-capacitor geometry of EFM induces backward-oriented and exterior field lines, making the induced generative transport stochastic and complicating practical ODE-based inference. To this end, we propose Capacitor Chain Field Matching (C$^2$FM), a generative modeling framework for data-to-data transport based on a periodic chain of electrostatic capacitors. Instead of using a single pair of charged plates, C$^2$FM places infinitely many alternating copies of the source and target distributions along the auxiliary dimension. This periodic construction creates a symmetry that eliminates flux leakage beyond adjacent source and target plates, thereby preventing backward-oriented trajectories. Consequently, samples can be transported deterministically by integrating a forward-oriented field from the source distribution to the target distribution. We prove that the induced field-line map transports the source distribution to the target distribution and provide a practical field-matching algorithm. Our framework preserves the electrostatic interpretation of EFM while yielding a geometrically well-behaved generative transport mechanism for data-to-data transfer.