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

Efficiently solving Fokker–Planck equations (FPEs) is crucial for understanding the probabilistic evolution of stochastic particles in dynamical systems, however, analytical solutions 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 probability solution (PAPS). Unlike previous numerical methodologies that necessitate solving the FPEs separately for each set of system parameters, the PAPS simultaneously addresses all 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 functions, namely, the solutions to FPEs. By leveraging a deep residual network, each parameter configuration of the system is mapped to the parameters of the GMD, ensuring that the weights, means, and variances of 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 PAPS obeying nonnegativity, normalization and boundary conditions. Extensive numerical studies validate the accuracy and efficiency of our method. This approach presents new insight to the pseudo-analytical solutions to FPEs, and promises significant acceleration in the response analysis of multi-parameter, multi-dimensional stochastic nonlinear systems.