A nonlocal physics-informed deep learning framework using the peridynamic differential operator

TL;DR

This paper introduces a nonlocal physics-informed neural network framework using the Peridynamic Differential Operator to better handle sharp gradients in PDE solutions, improving accuracy and parameter inference.

Contribution

It develops a novel nonlocal PINN approach incorporating long-range interactions via PDDO, enhancing performance over traditional local PINNs in problems with sharp gradients.

Findings

Nonlocal PINN outperforms local PINN in accuracy.

Nonlocal PINN improves parameter inference in solid mechanics.

Method effectively handles localized deformation with sharp gradients.

Abstract

The Physics-Informed Neural Network (PINN) framework introduced recently incorporates physics into deep learning, and offers a promising avenue for the solution of partial differential equations (PDEs) as well as identification of the equation parameters. The performance of existing PINN approaches, however, may degrade in the presence of sharp gradients, as a result of the inability of the network to capture the solution behavior globally. We posit that this shortcoming may be remedied by introducing long-range (nonlocal) interactions into the network's input, in addition to the short-range (local) space and time variables. Following this ansatz, here we develop a nonlocal PINN approach using the Peridynamic Differential Operator (PDDO)---a numerical method which incorporates long-range interactions and removes spatial derivatives in the governing equations. Because the PDDO functions can be readily incorporated in the neural network architecture, the nonlocality does not degrade the performance of modern deep-learning algorithms. We apply nonlocal PDDO-PINN to the solution and identification of material parameters in solid mechanics and, specifically, to elastoplastic deformation in a domain subjected to indentation by a rigid punch, for which the mixed displacement--traction boundary condition leads to localized deformation and sharp gradients in the solution. We document the superior behavior of nonlocal PINN with respect to local PINN in both solution accuracy and parameter inference, illustrating its potential for simulation and discovery of partial differential equations whose solution develops sharp gradients.