Using reservoir computing to solve FPK equations for stochastic dynamical systems under Gaussian or Non-Gaussian excitation

Abstract

This paper presents a new approach that uses the Reservoir Computing Algorithm to solve Fokker-Planck-Kolmogorov (FPK) equation excited by both Gaussian white noise and non-Gaussian noise. Unlike typical numerical methods, this methodology does not necessitate spatial reconstruction or numerical supplementation. The novelty of this paper lies in the modifications made to the conventional Reservoir Computing algorithm. We altered the approach for calculating values of the input weight matrix and incorporated autoregressive techniques in the reservoir layer. In addition, we applied data normalization to the training data before training the algorithm to avoid a zero solution. The efficacy of this approach was verified through multiple arithmetic examples, showcasing its practicality and efficiency in solving FPK equations. Moreover, the Reservoir Computing-FPK algorithm is capable of solving high-dimensional and fractional-order FPK equations with a smaller training set than earlier algorithms. Finally, we analyzed how values of the input weight matrix and regularization parameter affected the performance of the algorithm. The findings suggest that the careful selection of hyperparameters can greatly improve the performance of the Reservoir Computing algorithm.