Physics Informed Neural Networks for an Inverse Problem in Peridynamic Models

TL;DR

This paper introduces RBF-iPINN, a physics-informed neural network approach using radial basis functions to solve inverse problems in peridynamic models, demonstrating improved physical consistency and accuracy.

Contribution

The paper proposes integrating radial basis functions into PINNs specifically for inverse peridynamic problems, highlighting the importance of RBF selection for meaningful solutions.

Findings

RBF-iPINN accurately recovers the peridynamic kernel

Solutions align well with physical expectations

Numerical experiments validate the approach

Abstract

Deep learning is a powerful tool for solving data driven differential problems and has come out to have successful applications in solving direct and inverse problems described by PDEs, even in presence of integral terms. In this paper, we propose to apply radial basis functions (RBFs) as activation functions in suitably designed Physics Informed Neural Networks (PINNs) to solve the inverse problem of computing the peridynamic kernel in the nonlocal formulation of classical wave equation, resulting in what we call RBF-iPINN. We show that the selection of an RBF is necessary to achieve meaningful solutions, that agree with the physical expectations carried by the data. We support our results with numerical examples and experiments, comparing the solution obtained with the proposed RBF-iPINN to the exact solutions.