A mixed formulation for physics-informed neural networks as a potential\n solver for engineering problems in heterogeneous domains: comparison with\n finite element method

Abstract

Physics-informed neural networks (PINNs) are capable of finding the solution\nfor a given boundary value problem. We employ several ideas from the finite\nelement method (FEM) to enhance the performance of existing PINNs in\nengineering problems. The main contribution of the current work is to promote\nusing the spatial gradient of the primary variable as an output from separated\nneural networks. Later on, the strong form which has a higher order of\nderivatives is applied to the spatial gradients of the primary variable as the\nphysical constraint. In addition, the so-called energy form of the problem is\napplied to the primary variable as an additional constraint for training. The\nproposed approach only required up to first-order derivatives to construct the\nphysical loss functions. We discuss why this point is beneficial through\nvarious comparisons between different models. The mixed formulation-based PINNs\nand FE methods share some similarities. While the former minimizes the PDE and\nits energy form at given collocation points utilizing a complex nonlinear\ninterpolation through a neural network, the latter does the same at element\nnodes with the help of shape functions. We focus on heterogeneous solids to\nshow the capability of deep learning for predicting the solution in a complex\nenvironment under different boundary conditions. The performance of the\nproposed PINN model is checked against the solution from FEM on two prototype\nproblems: elasticity and the Poisson equation (steady-state diffusion problem).\nWe concluded that by properly designing the network architecture in PINN, the\ndeep learning model has the potential to solve the unknowns in a heterogeneous\ndomain without any available initial data from other sources. Finally,\ndiscussions are provided on the combination of PINN and FEM for a fast and\naccurate design of composite materials in future developments.\n