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

arXiv: https://arxiv.org/abs/2006.00446 · HTML: https://mlsanity.com/paper/2006.00446/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 May 2020.

## Built on by (most cited first, 21 of 21)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method](https://mlsanity.com/paper/2110.15832/) (2021)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training](https://mlsanity.com/paper/2110.03049/) (2021)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Constitutive model characterization and discovery using physics-informed deep learning](https://mlsanity.com/paper/2203.09789/) (2022)
- [A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity](https://mlsanity.com/paper/2108.07243/) (2021)
- [Physics-informed neural network solution of thermo-hydro-mechanical (THM) processes in porous media](https://mlsanity.com/paper/2203.01514/) (2022)
- [An unsupervised latent/output physics-informed convolutional-LSTM network for solving partial differential equations using peridynamic differential operator](https://mlsanity.com/paper/2210.12177/) (2022)
- [Component Fourier Neural Operator for Singularly Perturbed Differential Equations](https://mlsanity.com/paper/2409.04779/) (2024)
- [Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics](https://mlsanity.com/paper/2311.12915/) (2023)
- [Deep learning for solution and inversion of structural mechanics and vibrations](https://mlsanity.com/paper/2105.09477/) (2021)
- [Physics-aware deep learning framework for linear elasticity](https://mlsanity.com/paper/2302.09668/) (2023)
- [Utilising physics-guided deep learning to overcome data scarcity](https://mlsanity.com/paper/2211.15664/) (2022)
- [Physics-informed radial basis network (PIRBN): A local approximating neural network for solving nonlinear PDEs](https://mlsanity.com/paper/2304.06234/) (2023)
- [A peridynamic-informed deep learning model for brittle damage prediction](https://mlsanity.com/paper/2310.01350/) (2023)
- [ML-based identification of the interface regions for coupling local and nonlocal models](https://mlsanity.com/paper/2404.15388/) (2024)
- [Predicting Stress in Two-phase Random Materials and Super-Resolution Method for Stress Images by Embedding Physical Information](https://mlsanity.com/paper/2504.18854/) (2025)
- [A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization](https://mlsanity.com/paper/2604.21411/) (2026)
- [Modeling Unknown Nonlocal PDE Systems via Flow Map Learning](https://mlsanity.com/paper/2608.00400/) (2026)

## Abstract (arXiv)

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.