Explainable machine learning with Fredholm Neural Networks

Kyriakos Georgiou, Constantinos Siettos, Athanasios Yannacopoulos

Keywords: Explainable machine learning, Numerical analysis, Deep neural networks, Fixed point iterations, Inverse problems

TL;DR: Based on a connection between neural networks and fixed point iterations, we develop a novel architecture for Deep Neural Networks with applications in both forward and inverse problems, AI for Science

Abstract:

Applications of FIEs include the solution of ordinary and partial differential equations (ODEs, PDEs) as well the solution of inverse problems. Within the family of explainable machine learning techniques, 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 first prove that Fredholm NNs provide accurate solutions. We then provide a rigorous approach for determining the values of the hyperparameters and trainable/explainable weights and biases of the DNN, by directly connecting their values to the underlying mathematical theory. We illustrate the applicability and performance of the proposed Fredholm NNs to the solution of forward and inverse linear and nonlinear problems, including elliptic PDEs and boundary value problems. We show that the proposed scheme achieves a significant numerical approximation accuracy across both the domain and the 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 and fruitful cross pollination between scientific machine learning and numerical analysis.

(https://arxiv.org/abs/2408.09484) - this work is also in the second stage of peer review for journal publication

Submission Number: 90