A physics-informed learning approach to Bernoulli-type free boundary problems

Abstract

• A PINN methodology for a Bernoulli free boundary variational problem is proposed. • Quasi Monte-Carlo integration methods are used in the learning process of the PINN. • A redistribution strategy of domain points based on solution's Laplacian is developed. • Analysis on convex and non-convex domains assess the PINN reliability. Bernoulli free boundary problems (BFBP) govern many real applications, from chemistry to fluid dynamics. BFBP are overdetermined differential models in which the boundary of the solution's domain appears as an unknown. This work provides a variational formulation for BFBP, and a computational method focused on the novel methodology of Physics Informed Neural Networks (PINNs) is proposed. It consists in training a neural network to approximate the solution of the differential problem, minimizing a suitable cost function built, taking into account the physics constraint given by the model. In particular, since physics laws are injected through a loss function containing integral terms in our case, an approach based on quasi Monte-Carlo integration methods has been implemented. Moreover, an adaptive strategy to increase the model accuracy, exploiting topological information related to the shape of the solution, has been developed. Finally, when available, comparisons with the analytical solutions and studies in the case of non-convex domains assess the reliability of the PINN approach in the examined context.