DeepWKB: Learning WKB Expansions of Invariant Distributions for Stochastic Systems

• URL: http://arxiv.org/abs/2508.09529v1
• Date: Wed, 13 Aug 2025 06:23:06 GMT
• Title: DeepWKB: Learning WKB Expansions of Invariant Distributions for Stochastic Systems
• Abstract summary: This paper introduces a novel deep learning method, called DeepWKB, for estimating the invariant distribution of randomly perturbed systems.<n>By utilizing both Monte Carlo data and the partial differential equations satisfied by $V$ and $Z_epsilon$, the DeepWKB method computes $V$ and $Z_epsilon$ separately.<n>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 stability of complex systems.
• Score: 6.482696821807494
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $u_\epsilon(x) = Q(\epsilon)^{-1} Z_\epsilon(x) \exp\{-V(x)/\epsilon\}$, where $V$ is known as the quasi-potential, $\epsilon$ denotes the noise strength, and $Q(\epsilon)$ is the normalization factor. By utilizing both Monte Carlo data and the partial differential equations satisfied by $V$ and $Z_\epsilon$, the DeepWKB method computes $V$ and $Z_\epsilon$ separately. This enables an approximation of the invariant distribution in the singular regime where $\epsilon$ 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.

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