Solving Bivariate Kinetic Equations for Polymer Diffusion Using Deep Learning

Abstract

In this paper, we derive a class of backward stochastic differential equations (BSDEs) for infinite-dimensionally coupled nonlinear parabolic partial differential equations, thereby extending the deep BSDE method. In addition, we introduce a class of polymer dynamics models that accompany polymerization and depolymerization reactions, and derive the corresponding Fokker-Planck equations and Feynman-Kac equations. Due to chemical reactions, the system exhibits a Brownian yet non-Gaussian phenomenon, and the resulting equations are infinitely dimensionally coupled. We solve these equations numerically through our new deep BSDE method, and also solve a class of high-dimensional nonlinear equations, which verifies the effectiveness and shows approximation accuracy of the algorithm.