  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.
