  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.
