The Fokker-Planck (FP) equation is a linear partial differential equation which governs the temporal and spatial evolution of the probability density function (PDF) associated with the response of stochastic dynamical systems. An exact analytical solution of the FP equation is only available for a limited subset of dynamical systems. Semi-analytical methods are available for larger, yet still a small subset of systems, while traditional computational methods; e.g. Finite Elements and Finite Difference require dividing the computational domain into a grid of discrete points, which incurs significant computational costs for high-dimensional systems. Physics-informed learning offers a potentially powerful alternative to traditional computational schemes. To evaluate its potential, we present a data-free, physics-informed neural network (PINN) framework to solve the FP equation for a class of nonlinear stochastic dynamical systems. In particular, through several examples concerning the stochastic response of the Duffing, Van der Pol, and the Duffing-Van der Pol oscillators, we assess the ability and accuracy of the PINN framework in $i)$ predicting the PDF under the combined effect of additive and multiplicative noise, $ii)$ capturing P-bifurcations of the PDF, and $iii)$ effectively treating high-dimensional systems. Through comparisons with Monte-Carlo simulations and the available literature, we show that PINN can effectively address all of the afore-described points. We also demonstrate that the computational time associated with the PINN solution can be substantially reduced by using transfer learning.

通过物理信息神经网络（PINN）框架，解决一类非线性随机动力学系统的Fokker-Planck方程，通过Duffing、Van der Pol和Duffing-Van der Pol振子的几个示例评估PINN在预测PDF、捕捉PDF的P分岔和处理高维系统方面的能力和准确性，并通过与蒙特卡洛模拟和现有文献的比较证明PINN可以有效应对所有上述问题，同时表明使用迁移学习可以大大减少PINN解决方案的计算时间。

一类非线性动力系统的基于物理的解决方案评估研究中的静态傅克-普朗克方程