The pseudo-analytical density solution to parameterized Fokker-Planck equations via deep learning

Abstract

Efficiently solving the Fokker-Planck equation (FPE) is crucial for understanding the probabilistic evolution of stochastic particles in dynamical systems, however, analytical solutions or density functions are only attainable in specific cases. To speed up the solving process of parameterized FPEs with several system parameters, we introduce a deep learning-based method to obtain the pseudo-analytical density (PAD). Unlike previous numerical methodologies that necessitate solving the FPE separately for each set of system parameters, the PAD simultaneously addresses all the FPEs within a predefined continuous range of system parameters during a single training phase. The approach utilizes a Gaussian mixture distribution (GMD) to represent the stationary probability density, the solution to the FPE. By leveraging a deep residual network, each system parameter configuration is mapped to the parameters of the GMD, ensuring that the weights, means, and variances of the Gaussian components adaptively align with the corresponding true density functions. A grid-free algorithm is further developed to effectively train the residual network, resulting in a feasible PAD obeying necessary normalization and boundary conditions. Extensive numerical studies validate the accuracy and efficiency of our method, promising significant acceleration in the response analysis of multi-parameter, multi-dimensional stochastic nonlinear systems.