Nonlocal physics-informed neural networks – A unified theoretical and computational framework for nonlocal models

Nonlocal models provide an improved predictive capability thanks to their ability to capture effects that classical partial differential equations fail to capture. Among these effects we have multiscale behavior and anomalous behavior such as super- and sub-diffusion. These models have become incredibly popular for a broad range of applications, including mechanics, subsurface flow, turbulence, plasma dynamics, heat conduction and image processing. However, their improved accuracy comes at a price of many modeling and numerical challenges. In this work we focus on the estimation of model parameters, often unknown, or subject to noise. In particular, we address the problem of model identification in presence of sparse measurements. Our approach to this inverse problem is based on the combination of 1. Machine Learning and Physical Principles and 2. a Unified Nonlocal Vector Calculus and Versatile Surrogates such as neural networks (NN). The outcome is a flexible tool that allows us to learn existing and new nonlocal operators. We refer to our technique as nPINNs (nonlocal Physics-Informed Neural Networks); here, we model the nonlocal solution with a NN and we solve an optimization problem where we minimize the residual of the nonlocal equation and the misfit with measured data. The result of the optimization are the weights and biases of the NN and the set of unknown model parameters.