Fredholm Neural Networks for forward and inverse problems in elliptic PDEs

TL;DR

This paper introduces Potential Fredholm Neural Networks (PFNNs), an explainable deep learning framework for solving forward and inverse elliptic PDEs using boundary integral methods, with explicit error bounds and boundary condition adherence.

Contribution

The paper extends Fredholm Neural Networks to elliptic PDEs, providing an explainable, boundary-respecting neural network architecture with explicit error bounds for forward and inverse problems.

Findings

Achieves small interior errors and near machine-precision boundary accuracy.

Provides explicit error bounds based on boundary function approximation and discretization.

Demonstrates effectiveness for 2D linear and semi-linear elliptic PDEs.

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.