Neural Green's Function Accelerated Iterative Methods for Solving Indefinite Boundary Value Problems

TL;DR

This paper introduces a physics-informed, data-free neural operator that learns Green's functions directly to accelerate iterative methods for solving indefinite boundary value problems, especially with discontinuous coefficients.

Contribution

It presents a novel neural operator that learns Green's functions without data, reformulates the problem into an interface form, and develops a hybrid iterative solver leveraging spectral bias.

Findings

Accelerates convergence of iterative methods for indefinite problems.

Effective with discontinuous coefficients.

Theoretical and numerical validation of spectral bias effects.

Abstract

Neural operators, which learn mappings between the function spaces, have been applied to solve boundary value problems in various ways, including learning mappings from the space of the forcing terms to the space of the solutions with the substantial requirements of data pairs. In this work, we present a data-free neural operator integrated with physics, which learns the Green kernel directly. Our method proceeds in three steps: 1. The governing equations for the Green's function are reformulated into an interface problem, where the delta Dirac function is removed; 2. The interface problem is embedded in a lifted space of higher-dimension to handle the jump in the derivative, but still solved on a two-dimensional surface without additional sampling cost; 3. Deep neural networks are employed to address the curse of dimensionality caused by this lifting operation. The approximate Green's function obtained through our approach is then used to construct preconditioners for the linear systems allowed by its mathematical properties. Furthermore, the spectral bias of it revealed through both theoretical analysis and numerical validation contrasts with the smoothing effects of traditional iterative solvers, which motivates us to propose a hybrid iterative method that combines these two solvers. Numerical experiments demonstrate the effectiveness of our approximate Green's function in accelerating iterative methods, proving fast convergence for solving indefinite problems even involving discontinuous coefficients.