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

arXiv: https://arxiv.org/abs/2408.12389 · HTML: https://mlsanity.com/paper/2408.12389/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 22 references that are in the map.
- First posted on arXiv 20 Aug 2024.

## Builds on (most cited first, 10 of 10)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [BENO: Boundary-embedded Neural Operators for Elliptic PDEs](https://mlsanity.com/paper/2401.09323/) (2024)
- [Learning the Solution Operator of Boundary Value Problems using Graph Neural Networks](https://mlsanity.com/paper/2206.14092/) (2022)
- [Learning Only On Boundaries: a Physics-Informed Neural operator for Solving Parametric Partial Differential Equations in Complex Geometries](https://mlsanity.com/paper/2308.12939/) (2023)
- [Improving physics-informed DeepONets with hard constraints](https://mlsanity.com/paper/2309.07899/) (2023)

## Abstract (arXiv)

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.