DeepWKB: Learning WKB Expansions of Invariant Distributions for Stochastic Systems
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913988066934784 |
|---|---|
| author | Li, Yao Liu, Yicheng Wang, Shirou |
| author_facet | Li, Yao Liu, Yicheng Wang, Shirou |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09529 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | DeepWKB: Learning WKB Expansions of Invariant Distributions for Stochastic Systems Li, Yao Liu, Yicheng Wang, Shirou Dynamical Systems Machine Learning 60F10, 60J25, 37M25 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. |
| title | DeepWKB: Learning WKB Expansions of Invariant Distributions for Stochastic Systems |
| topic | Dynamical Systems Machine Learning 60F10, 60J25, 37M25 |
| url | https://arxiv.org/abs/2508.09529 |