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.

翻译：我们将保持稳态密度不变的非互易Fokker-Planck动力学微扰表述为规范场，该类规范场在保持不变态固定的同时使弛豫谱发生形变。当细致平衡成立时，相似变换将可逆Fokker-Planck算符映射为Witten-Laplacian型超对称哈密顿量；非互易规范则作为非厄米微扰出现，其保持零模但修改激发谱。该算符视角为弛豫隙、循环概率流、低余切加速及有限控制代价提供了统一语言。我们通过反对称张量场表示可容许规范流，并将违反细致平衡的Ohzeki-Ichiki力识别为常辛流形示例，其无穷强度极限退化为哈密顿动力学。由于连续时间谱隙本身无法唯一选取有限规范强度，我们引入有限时间正则化目标及演员-评论家学习程序以习得规范。通过精确可解的异质高斯Ornstein-Uhlenbeck基准实验，我们分离了谱跃迁与有限时间最优解，并证明习得规范可恢复Lyapunov方程最优解。双势阱基准实验进一步展示了非凸亚稳态景观中的相同约束选择问题。随机梯度方法在此框架中作为具有物理相关性的Fokker-Planck系统自然呈现：小批量噪声等效于有效扩散张量，而Adam等自适应方法对应于可能包含非平衡流的度量选择。