用深度学习估算弱噪声下系统的稳态分布，突破传统方法极限。
DeepWKB: Learning WKB Expansions of Invariant Distributions for Stochastic Systems
- 结合蒙特卡洛数据与偏微分方程，分步求解势能与归一化因子。
- 在小噪声条件下成功逼近高维系统稳态分布，精度显著提升。
- 适用于复杂系统罕见事件分析，适合做随机动力学研究者参考。
本文提出一种新型深度学习方法 DeepWKB，通过 WKB 近似 $u_ε(x) = Q(ε)^{-1} Z_ε(x) \ exp\{-V(x)/ε\}$ 估算随机扰动系统的不变分布，其中 $V$ 为拟势，$ε$ 为噪声强度，$Q(ε)$ 为归一化因子。DeepWKB 利用蒙特卡洛数据与 $V$、$Z_ε$ 满足的偏微分方程，分别计算二者，从而在 $ε$ 足够小时实现对不变分布的逼近，克服了现有方法在奇异极限下的难题。该方法可推广至具有非平凡吸引子的高维随机系统，为稀有事件、亚稳态及复杂系统随机稳定性分析中的拟势计算提供了可扩展且灵活的替代方案。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。