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

arXiv: https://arxiv.org/abs/2606.06412 · HTML: https://mlsanity.com/paper/2606.06412/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 54 references that are in the map.
- First posted on arXiv 4 Jun 2026.

## Builds on (most cited first, 28 of 28)

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- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Symbolic Discovery of Optimization Algorithms](https://mlsanity.com/paper/2302.06675/) (2023)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
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- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
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- [Improving Generalization Performance by Switching from Adam to SGD](https://mlsanity.com/paper/1712.07628/) (2017)
- [On the Generalization Benefit of Noise in Stochastic Gradient Descent](https://mlsanity.com/paper/2006.15081/) (2020)
- [Generalizing Hamiltonian Monte Carlo with Neural Networks](https://mlsanity.com/paper/1711.09268/) (2017)
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- [The effective noise of Stochastic Gradient Descent](https://mlsanity.com/paper/2112.10852/) (2021)
- [Stochastic gradient descent introduces an effective landscape-dependent regularization favoring flat solutions](https://mlsanity.com/paper/2206.01246/) (2022)
- [Improving the convergence of reversible samplers](https://mlsanity.com/paper/1601.08118/) (2016)
- [On Large Batch Training and Sharp Minima: A Fokker-Planck Perspective](https://mlsanity.com/paper/2112.00987/) (2021)
- [Mathematical understanding of detailed balance condition violation and its application to Langevin dynamics](https://mlsanity.com/paper/1503.02356/) (2015)
- [Stochastic gradient method with accelerated stochastic dynamics](https://mlsanity.com/paper/1511.06036/) (2015)
- [SGD at the Edge of Stability: The Stochastic Sharpness Gap](https://mlsanity.com/paper/2604.21016/) (2026)

## Abstract (arXiv)

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.