The invariant distribution, which is characterized by the stationary Fokker-Planck equation, is an important object in the study of randomly perturbed dynamical systems. Traditional numerical methods for computing the invariant distribution based on the Fokker-Planck equation, such as finite difference or finite element methods, are limited to low-dimensional systems due to the curse of dimensionality. In this work, we propose a deep learning based method to compute the generalized potential, i.e. the negative logarithm of the invariant distribution multiplied by the noise. The idea of the method is to learn a decomposition of the force field, as specified by the Fokker-Planck equation, from the trajectory data. The potential component of the decomposition gives the generalized potential. The method can deal with high-dimensional systems, possibly with partially known dynamics. Using the generalized potential also allows us to deal with systems at low temperatures, where the invariant distribution becomes singular around the metastable states. These advantages make it an efficient method to analyze invariant distributions for practical dynamical systems. The effectiveness of the proposed method is demonstrated by numerical examples.

翻译：静态 Fokker- Planck 等式是随机扰动动态系统研究的一个重要对象。基于 Fokker-Planck 等式计算异质分布的传统数字方法,例如定分法或定分法,由于维度的诅咒,仅限于低维系统。在这项工作中,我们建议一种深层次的基于学习的方法,以计算普遍潜力,即静态Fokker-Planck 等式的负对数分布乘以噪音。该方法的构想是从轨迹数据中学习Fokker-Planck 等式所指定的动力场的分解方法。分解法的潜在组成部分提供了通用潜力。该方法可以处理高维系统,可能部分具有已知的动态。使用普遍潜力还使我们能够处理低温系统,即逆差分布在元数状态周围变得单数。这些优势使得它成为分析实际动态系统变量分布的有效方法。拟议方法的有效性通过数字示例展示。