Physics Informed Neural Networks for Learning the Horizon Size in Bond-Based Peridynamic Models

TL;DR

This paper uses Physics-Informed Neural Networks to solve the inverse problem of estimating the horizon size in bond-based peridynamic models, demonstrating effectiveness across various kernel functions and dimensions.

Contribution

It introduces PINNs for horizon size estimation in peridynamics and analyzes the convergence behavior of the training process.

Findings

PINNs successfully learn the horizon parameter in 1D and 2D models.

The method is effective even with complex kernel functions.

SGD exhibits one-sided convergence towards a global minimum.

Abstract

This paper broaches the peridynamic inverse problem of determining the horizon size of the kernel function in a one-dimensional model of a linear microelastic material. We explore different kernel functions, including V-shaped, distributed, and tent kernels. The paper presents numerical experiments using PINNs to learn the horizon parameter for problems in one and two spatial dimensions. The results demonstrate the effectiveness of PINNs in solving the peridynamic inverse problem, even in the presence of challenging kernel functions. We observe and prove a one-sided convergence behavior of the Stochastic Gradient Descent method towards a global minimum of the loss function, suggesting that the true value of the horizon parameter is an unstable equilibrium point for the PINN's gradient flow dynamics.