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

TL;DR

This work addresses data-driven boundary value problems on irregular domains where governing equations may be unknown. It proposes FIE-NO, a physics-guided neural operator that embeds Fredholm Integral Equations into a neural architecture, leveraging Random Fourier Features to approximate kernels and two parallel networks (KAN and IAN) to learn the kernel and integral components. The method extends to Neumann conditions and demonstrates superior accuracy across 2D Laplace, Helmholtz, and Darcy flow problems compared with NIE, ANIE, Green Operator, and Green Learning, while training on a single boundary condition and generalizing to new boundaries. The approach offers a scalable, robust tool for solving complex BVPs with irregular geometries in computational physics and engineering.

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.

Figures (8)

• Figure 1: Model Structure: The diagram shows the overall model architecture, including the processing of boundary and interior points. Boundary points are processed through CNN to extract features, which are then transformed by MLP. Interior points are processed through ELM-inspired layers, followed by MLP with cosine activation and GELU activation. IAN and KAN handle integral and kernel approximations, with their outputs multiplied and summed to produce values for the interior points. For each training session, $m$ = 200 boundary points are sampled along the boundary, including boundary values $\varphi(x)$, Cartesian coordinates $(x, y)$, and polar coordinates $(r, \theta)$. These inputs are processed together through a Convolutional Neural Network (CNN) to extract features, which are then transformed by a MLP. Interior points include only their Cartesian and polar coordinates $(x, y)$ and $(r, \theta)$. These coordinates are processed through multiple Extreme Learning Machine (ELM)-inspired hidden layers with fixed weights, followed by a MLP network with cosine activation function and a learnable GELU activation function. The Integral Approximation Network (IAN) and Kernel Approximation Network (KAN) handle the integral and kernel approximations. Their outputs are multiplied and summed to produce the values for the interior points.
• Figure 2: Boundary
• Figure 3: Dirichlet Boundary conditions
• Figure 4: Neumann Boundary conditions
• Figure 5: Results