Solving Forward and Inverse Problems of Contact Mechanics using
  Physics-Informed Neural Networks
T. Sahin M. von Danwitz A. Popp
This paper explores the ability of physics-informed neural networks (PINNs)
to solve forward and inverse problems of contact mechanics for small
deformation elasticity. We deploy PINNs in a mixed-variable formulation
enhanced by output transformation to enforce Dirichlet and Neumann boundary
conditions as hard constraints. Inequality constraints of contact problems,
namely Karush-Kuhn-Tucker (KKT) type conditions, are enforced as soft
constraints by incorporating them into the loss function during network
training. To formulate the loss function contribution of KKT constraints,
existing approaches applied to elastoplasticity problems are investigated and
we explore a nonlinear complementarity problem (NCP) function, namely
Fischer-Burmeister, which possesses advantageous characteristics in terms of
optimization. Based on the Hertzian contact problem, we show that PINNs can
serve as pure partial differential equation (PDE) solver, as data-enhanced
forward model, as inverse solver for parameter identification, and as
fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of
choosing proper hyperparameters, e.g. loss weights, and a combination of Adam
and L-BFGS-B optimizers aiming for better results in terms of accuracy and
training time.