Fredholm Neural Networks

TL;DR

Fredholm neural networks are a new explainable deep learning framework inspired by integral equation solution methods, capable of accurately solving linear and nonlinear problems including PDEs, with potential applications in UQ and XAI.

Contribution

This paper introduces Fredholm neural networks, a novel DNN architecture grounded in integral equation theory, linking neural networks with classical numerical methods for scientific computing.

Findings

Achieves high numerical accuracy in solving PDEs and boundary value problems.

Provides interpretability by connecting network parameters to mathematical theory.

Demonstrates potential for applications in UQ and XAI.

Abstract

Within the family of explainable machine-learning, we present Fredholm neural networks (Fredholm NNs): deep neural networks (DNNs) architectures motivated by fixed-point iteration schemes for the solution of linear and nonlinear Fredholm integral equations (FIEs) of the second kind. We also show how the proposed framework can be used for the solution of inverse problems. Applications of FIEs include the solution of ordinary, as well as partial differential equations (ODEs, PDEs) and many more. We first prove that Fredholm NNs provide accurate solutions. We then provide insight into the values of the hyperparameters and trainable/explainable weights and biases of the DNN, by directly connecting their values to the underlying mathematical theory. For our illustrations, we use Fredholm NNs to solve both linear and nonlinear problems, including elliptic PDEs and boundary value problems. We show that the proposed scheme achieves significant numerical approximation accuracy across both the domain and boundary. The proposed methodology provides insight into the connection between neural networks and classical numerical methods, and we posit that it can have applications in fields such as Uncertainty Quantification (UQ) and explainable artificial intelligence (XAI). Thus, we believe that it will trigger further advances in the intersection between scientific machine learning and numerical analysis.