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

In this paper, we present a novel Fredholm Integral Equations Neural Operator (FIE-NO) method, an integration of Random Fourier Features (RFF) and Fredholm Integral Equations (FIE) into the deep learning framework, tailored for solving data-driven Boundary Value Problems (BVPs) with a particular focus on challenges posed by 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. By harnessing the power of physics-guided operator learning, FIE-NO demonstrates superior performance in addressing BVPs. Notably, our approach is designed to generalize across multiple scenarios, including those with unknown equation forms and intricate boundary shapes, after being trained on a singular boundary condition type. Experimental validation demonstrates that the FIE-NO method performs well in various fluid mechanics scenarios. Utilizing the Darcy flow equation and typical PDEs such as the Laplace and Helmholtz equations, the 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.