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

Bernoulli free boundary problems (BFBP) govern many real applications, from chemistry to fluid dynamics. 89 BFBP are overdetermined differential models in which the boundary of the solution’s domain appears as an 90 unknown. This work provides a variational formulation for BFBP, and a computational method focused on the 91 novel methodology of Physics Informed Neural Networks (PINNs) is proposed. It consists in training a neural 92 network to approximate the solution of the differential problem, minimizing a suitable cost function built, taking 93 into account the physics constraint given by the model. In particular, since physics laws are injected through a 94 loss function containing integral terms in our case, an approach based on quasi Monte-Carlo integration methods 95 has been implemented. Moreover, an adaptive strategy to increase the model accuracy, exploiting topological 96 information related to the shape of the solution, has been developed. Finally, when available, comparisons with 97 the analytical solutions and studies in the case of non-convex domains assess the reliability of the PINN approach 98 in the examined context.