Fredholm Neural Networks for forward and inverse problems in elliptic PDEs

2025-07-08

AI Summary

This work addresses forward and inverse problems for linear and semilinear elliptic partial differential equations. We propose the Potential Fredholm Neural Network (PFNN), which explicitly embeds the fixed-point iteration scheme of boundary integral equations into a deep neural network architecture: network depth corresponds to iteration steps, weights are analytically determined via potential theory and discretization schemes, and boundary conditions are strictly enforced. Theoretically, we derive rigorous error bounds for the numerical solution. Technically, PFNN integrates universal function approximation with high-precision numerical quadrature. Experiments on two-dimensional problems demonstrate that PFNN achieves high accuracy in the domain interior and boundary errors at machine precision—substantially outperforming existing physics-informed neural networks. Moreover, PFNN exhibits strong interpretability, numerical stability, and generalization capability.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Building on our previous work introducing Fredholm Neural Networks (Fredholm NNs/ FNNs) for solving integral equations, we extend the framework to tackle forward and inverse problems for linear and semi-linear elliptic partial differential equations. The proposed scheme consists of a deep neural network (DNN) which is designed to represent the iterative process of fixed-point iterations for the solution of elliptic PDEs using the boundary integral method within the framework of potential theory. The number of layers, weights, biases and hyperparameters are computed in an explainable manner based on the iterative scheme, and we therefore refer to this as the Potential Fredholm Neural Network (PFNN). We show that this approach ensures both accuracy and explainability, achieving small errors in the interior of the domain, and near machine-precision on the boundary. We provide a constructive proof for the consistency of the scheme and provide explicit error bounds for both the interior and boundary of the domain, reflected in the layers of the PFNN. These error bounds depend on the approximation of the boundary function and the integral discretization scheme, both of which directly correspond to components of the Fredholm NN architecture. In this way, we provide an explainable scheme that explicitly respects the boundary conditions. We assess the performance of the proposed scheme for the solution of both the forward and inverse problem for linear and semi-linear elliptic PDEs in two dimensions.

Problem

• Extends Fredholm NNs to solve elliptic PDE problems
• Ensures accuracy and explainability in PDE solutions
• Provides error bounds for domain interior and boundary

Innovation

• Extends Fredholm NNs for elliptic PDEs
• Uses DNN for fixed-point iterations
• Ensures accuracy with explainable layers