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.

最近引入的物理信息神经网络 (PINN) 框架将物理学纳入深度学习，并为偏微分方程(PDE) 的求解以及方程参数的识别提供了一条有前途的途径。然而，由于网络无法全局捕获解决方案行为，现有 PINN 方法的性能可能会在存在陡峭梯度的情况下降低。我们假设除了短程（本地）空间和时间变量之外，还可以通过在网络输入中引入长程（非本地）交互来弥补这一缺点。在此 ansatz 之后，我们在这里开发了一种使用近场动力学微分算子的非局部PINN 方法(PDDO)——一种数值方法，它结合了长程相互作用并去除了控制方程中的空间导数。因为 PDDO 函数可以很容易地合并到神经网络架构中，非局部性不会降低现代深度学习算法的性能。我们将非局部 PDDO-PINN 应用于固体力学中材料参数的求解和识别，特别是在受到刚性冲头压痕的域中的弹塑性变形，为此混合位移-牵引边界条件导致局部变形和尖锐溶液中的梯度。我们记录了非局部 PINN 在解精度和参数推断方面相对于局部 PINN 的优越行为，说明了其模拟和发现偏微分方程的潜力，其解会产生陡峭的梯度。