标题： DeepWKB：学习随机系统的不变分布的WKB展开式 显示英文标题

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

摘要： 本文介绍了一种新颖的深度学习方法，称为DeepWKB，用于通过其Wentzel-Kramers-Brillouin（WKB）近似$u_\epsilon(x) = Q(\epsilon)^{-1} Z_\epsilon(x) \exp\{-V(x)/\epsilon\}$来估计随机扰动系统的不变分布，其中$V$称为拟势，$\epsilon$表示噪声强度，$Q(\epsilon)$是归一化因子。 通过同时利用蒙特卡洛数据以及$V$和$Z_\epsilon$所满足的偏微分方程，DeepWKB方法分别计算$V$和$Z_\epsilon$。 这使得在$\epsilon$足够小的奇异情形下能够对不变分布进行近似，而这对于大多数现有方法仍然是一个重大挑战。 此外，DeepWKB方法适用于其确定性对应物具有非平凡吸引子的高维随机系统。 特别是，它为计算准势提供了一个可扩展且灵活的替代方法，准势在罕见事件、亚稳态以及复杂系统的随机稳定性分析中起着关键作用。 显示英文摘要

摘要： 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.