A mixed formulation for physicsinformed neural networks as a potential solver for engineering problems in heterogeneous domains: Comparison with finite element method
Abstract
Physicsinformed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering problems. The main contribution of the current work is to promote using the spatial gradient of the primary variable as an output from separated neural networks. Later on, the strong form which has a higher order of derivatives is applied to the spatial gradients of the primary variable as the physical constraint. In addition, the socalled energy form of the problem is applied to the primary variable as an additional constraint for training. The proposed approach only required up to firstorder derivatives to construct the physical loss functions. We discuss why this point is beneficial through various comparisons between different models. The mixed formulationbased PINNs and FE methods share some similarities. While the former minimizes the PDE and its energy form at given collocation points utilizing a complex nonlinear interpolation through a neural network, the latter does the same at element nodes with the help of shape functions. We focus on heterogeneous solids to show the capability of deep learning for predicting the solution in a complex environment under different boundary conditions. The performance of the proposed PINN model is checked against the solution from FEM on two prototype problems: elasticity and the Poisson equation (steadystate diffusion problem). We concluded that by properly designing the network architecture in PINN, the deep learning model has the potential to solve the unknowns in a heterogeneous domain without any available initial data from other sources. Finally, discussions are provided on the combination of PINN and FEM for a fast and accurate design of composite materials in future developments.
 Publication:

Computer Methods in Applied Mechanics and Engineering
 Pub Date:
 November 2022
 DOI:
 10.1016/j.cma.2022.115616
 arXiv:
 arXiv:2206.13103
 Bibcode:
 2022CMAME.401k5616R
 Keywords:

 Computer Science  Computational Engineering;
 Finance;
 and Science;
 Computer Science  Machine Learning
 EPrint:
 doi:10.1016/j.cma.2022.115616