Effects of variational formulations on physics‐informed neural network performance in solid mechanics

PAMM, vol. 23

Abstract

Abstract In recent years, Physics‐Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) in a variety of fields. To this end, an artificial neural network is used to approximate the unknown variables while the differential equation is embedded in the loss function to penalize the network's output. The present contribution focuses on the application of PINNs and its derivatives in the field of solid mechanics. We formulate the PINN framework by incorporating the strong form of the corresponding PDEs alongside initial and boundary conditions into the loss function of the neural network. The loss function poses as a residual, effectively constructing a minimization problem to solve the PDEs. The nature of the residual is further reformulated into the variations (weak) form by applying test functions and integration by parts. This extension leads to Variational Physics‐Informed Neural Networks (VPINNs), which impose a lower stringency on the solution. We demonstrate the performance of PINNs and VPINNs on a numerical solid mechanics problem.

Authors 3

1. Nils Radin corresponding Aachen Chair of Structural Analysis and Dynamics

   RWTH Aachen University

   Affiliation as printed

   Chair of Structural Analysis and Dynamics RWTH Aachen University Aachen Germany

   Chair of Structural Analysis and Dynamics, RWTH Aachen University, Aachen, Germany

2. Sven Klinkel Aachen Chair of Structural Analysis and Dynamics

   RWTH Aachen University

   Affiliation as printed

   Chair of Structural Analysis and Dynamics RWTH Aachen University Aachen Germany

   Chair of Structural Analysis and Dynamics, RWTH Aachen University, Aachen, Germany

3. Okyay Altay Aachen Chair of Structural Analysis and Dynamics

   RWTH Aachen University

   Affiliation as printed

   Chair of Structural Analysis and Dynamics RWTH Aachen University Aachen Germany

   Chair of Structural Analysis and Dynamics, RWTH Aachen University, Aachen, Germany