An explainable operator approximation framework under the guideline of Green’s function

Abstract

Compared to traditional methods such as the finite element and finite volume methods, the Green’s function approach offers the advantage of providing analytical solutions to linear partial differential equations (PDEs) with varying boundary conditions and source terms, without the need for repeated iterative solutions. Nevertheless, deriving Green’s functions analytically remains a non-trivial task. In this study, we develop a framework inspired by the architecture of deep operator networks (DeepONet) to learn embedded Green’s functions and solve PDEs through integral formulation, termed the Green’s operator network (GON). Specifically, the Trunk Net within GON is designed to approximate the unknown Green’s functions of the system, while the Branch Net are utilized to approximate the auxiliary gradients of the Green’s function. These outputs are subsequently employed to perform surface integrals and volume integrals, incorporating user-defined boundary conditions and source terms, respectively. The effectiveness of the proposed framework is demonstrated on three types of PDEs in 3D bounded domains: Poisson equations, reaction-diffusion equations, and Stokes equations. Comparative results in these cases demonstrate that GON’s accuracy and generalization ability surpass those of existing methods, including Physics-Informed Neural Networks (PINN), DeepONet, Physics-Informed DeepONet (PI-DeepONet), and Fourier Neural Operators (FNO). Code and data is available at https://github.com/hangjianggu/GreensONet.