Fredholm Integral Equations Neural Operator (FIE-NO) for Data-Driven Boundary Value Problems

TL;DR

The paper introduces FIE-NO, a physics-inspired neural operator combining Fourier features and Fredholm equations, designed to efficiently solve complex boundary value problems with irregular boundaries, demonstrating superior accuracy and generalization.

Contribution

It presents a novel neural operator framework integrating Fredholm integral equations and Fourier features for data-driven boundary value problems with irregular boundaries.

Findings

FIE-NO outperforms existing methods in accuracy and stability.

It generalizes across different boundary conditions and equations.

Demonstrates effectiveness on Darcy, Laplace, and Helmholtz equations.

Abstract

In this paper, we present a novel Fredholm Integral Equation Neural Operator (FIE-NO) method, an integration of Random Fourier Features and Fredholm Integral Equations (FIE) into the deep learning framework, tailored for solving data-driven Boundary Value Problems (BVPs) with irregular boundaries. Unlike traditional computational approaches that struggle with the computational intensity and complexity of such problems, our method offers a robust, efficient, and accurate solution mechanism, using a physics inspired design of the learning structure. We demonstrate that the proposed physics-guided operator learning method (FIE-NO) achieves superior performance in addressing BVPs. Notably, our approach can generalize across multiple scenarios, including those with unknown equation forms and intricate boundary shapes, after being trained only on one boundary condition. Experimental validation demonstrates that the FIE-NO method performs well in simulated examples, including Darcy flow equation and typical partial differential equations such as the Laplace and Helmholtz equations. The proposed method exhibits robust performance across different boundary conditions. Experimental results indicate that FIE-NO achieves higher accuracy and stability compared to other methods when addressing complex boundary value problems with varying numbers of interior points.