# DeepWKB: Learning WKB Expansions of Invariant Distributions for Stochastic Systems

arXiv: https://arxiv.org/abs/2508.09529 · HTML: https://mlsanity.com/paper/2508.09529/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 47 references that are in the map.
- First posted on arXiv 13 Aug 2025.

## Builds on (most cited first, 2 of 2)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Data Driven Method for Computing Quasipotentials](https://mlsanity.com/paper/2012.09111/) (2020)

## Abstract (arXiv)

This paper introduces a novel deep learning method, called DeepWKB, for estimating the invariant distribution of randomly perturbed systems via its Wentzel-Kramers-Brillouin (WKB) approximation uε(x) = Q(ε)-1 Zε(x) exp-V(x)/ε, where V is known as the quasi-potential, ε denotes the noise strength, and Q(ε) is the normalization factor. By utilizing both Monte Carlo data and the partial differential equations satisfied by V and Zε, the DeepWKB method computes V and Zε separately. This enables an approximation of the invariant distribution in the singular regime where ε is sufficiently small, which remains a significant challenge for most existing methods. Moreover, the DeepWKB method is applicable to higher-dimensional stochastic systems whose deterministic counterparts admit non-trivial attractors. In particular, it provides a scalable and flexible alternative for computing the quasi-potential, which plays a key role in the analysis of rare events, metastability, and the stochastic stability of complex systems.