DeepWKB: Learning WKB Expansions of Invariant Distributions for Stochastic Systems
This provides a scalable alternative for analyzing rare events and metastability in complex stochastic systems, though it is incremental as it builds on existing WKB approximations.
The paper tackles the challenge of estimating invariant distributions for stochastic systems in the singular regime where noise strength is small, by introducing DeepWKB, a deep learning method that computes the quasi-potential and normalization factor separately, enabling scalable approximation for higher-dimensional systems with non-trivial attractors.
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 $u_ε(x) = Q(ε)^{-1} Z_ε(x) \exp\{-V(x)/ε\}$, 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 $Z_ε$, the DeepWKB method computes $V$ and $Z_ε$ 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.