Physics-Informed Solution of The Stationary Fokker-Plank Equation for a Class of Nonlinear Dynamical Systems: An Evaluation Study

Abstract

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.