Learning Green’s Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver

Yuankai Teng, Xiaoping Zhang, Zhu Wang, Lili Ju

Abstract: Partial differential equations are often used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and so on. Due to their important applications in scientific research and engineering, many numerical methods have been developed in the past decades for efficient and accurate solutions of these equations. Inspired by the rapidly growing impact of deep learning techniques, we propose in this paper a novel neural network method, “GF-Net”, for learning the Green’s functions of the classic linear reaction-diffusion equations in the unsupervised fashion. The proposed method overcomes the challenges for finding the Green’s functions of the equations on arbitrary domains by utilizing the physics-informed neural network approach and domain decomposition. As a consequence, it particularly leads to a fast algorithm for solving the target equations subject to various sources and Dirichlet boundary conditions without network retraining. We also numerically demonstrate the effectiveness of the proposed method by extensive experiments in the square, annular and L-shape domains.