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

TL;DR

GreensONet is a novel deep operator network framework that learns Green's functions to efficiently solve PDEs, outperforming existing neural network methods in accuracy and generalization across various equations.

Contribution

This work introduces GreensONet, a deep operator network that learns Green's functions for PDEs, enabling efficient and accurate solutions with improved generalization over existing methods.

Findings

GreensONet achieves higher accuracy than PINN, DeepONet, PI-DeepONet, and FNO.

GreensONet generalizes well to different PDEs and boundary conditions.

The framework effectively incorporates boundary conditions and source terms.

Abstract

Traditional numerical methods, such as the finite element method and finite volume method, adress partial differential equations (PDEs) by discretizing them into algebraic equations and solving these iteratively. However, this process is often computationally expensive and time-consuming. An alternative approach involves transforming PDEs into integral equations and solving them using Green's functions, which provide analytical solutions. Nevertheless, deriving Green's functions analytically is a challenging and non-trivial task, particularly for complex systems. In this study, we introduce a novel framework, termed GreensONet, which is constructed based on the strucutre of deep operator networks (DeepONet) to learn embedded Green's functions and solve PDEs via Green's integral formulation. Specifically, the Trunk Net within GreensONet 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 bounded domains: 3D heat conduction equations, reaction-diffusion equations, and Stokes equations. Comparative results in these cases demonstrate that GreenONet'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).