Nonreversible Gauge Fields in Fokker--Planck Dynamics: Supersymmetric Hamiltonians and Learned Finite Forces

2026-06-04

AI Summary

This work addresses the acceleration of convergence in Fokker–Planck dynamics under non-reversible perturbations while preserving the stationary distribution. It introduces a novel formulation by modeling such perturbations as gauge fields and unifying them within the framework of supersymmetric quantum mechanics, enabling spectral analysis of relaxation modes via the Witten Laplacian augmented with non-Hermitian gauge terms. To overcome limitations of traditional spectral gap optimization, the authors propose a finite-time regularized objective function and employ an actor-critic algorithm to learn the optimal gauge force. Numerical experiments on Gaussian Ornstein–Uhlenbeck and double-well potential models demonstrate that the learned gauge forces accurately recover theoretical optima, validating the method’s efficacy in both convex and non-convex landscapes. The study further establishes a physical correspondence between adaptive stochastic gradient methods and nonequilibrium Fokker–Planck systems.

Abstract

We formulate stationary-density-preserving nonreversible perturbations of Fokker--Planck dynamics as gauge fields that deform relaxation spectra while leaving the invariant state fixed. When detailed balance holds, a similarity transformation maps the reversible Fokker--Planck operator to a Witten-Laplacian-type supersymmetric Hamiltonian; nonreversible gauges then appear as non-Hermitian perturbations that preserve the zero mode but modify the excited spectrum. This operator viewpoint gives a common language for relaxation gaps, circulating probability currents, hypocoercive acceleration, and finite control costs. We represent admissible gauge currents by antisymmetric tensor fields and identify the detailed-balance-violating Ohzeki--Ichiki force as a constant symplectic example whose infinite-strength limit is Hamiltonian dynamics. The continuous-time spectral gap alone does not select a finite gauge strength, so we introduce a finite-time regularized objective and an actor--critic procedure for learning the gauge. An exactly solvable anisotropic Gaussian Ornstein--Uhlenbeck benchmark separates the spectral transition from the finite-time optimum and shows that the learned gauge recovers the Lyapunov-equation optimum. A double-well benchmark then illustrates the same constrained selection in a nonconvex metastable landscape. Stochastic gradient methods enter this framework as physically relevant Fokker--Planck systems: mini-batch noise acts as an effective diffusion tensor, and adaptive methods such as Adam correspond to metric choices with possible nonequilibrium currents.

Problem

• nonreversible perturbations
• Fokker-Planck dynamics
• stationary density preservation
• finite-time convergence
• spectral gap

Innovation

• nonreversible dynamics
• gauge fields
• supersymmetric Hamiltonian
• finite-time optimization
• Fokker-Planck learning