Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problems

Physics can often be described by partial differential equations (PDEs) with suitable boundary conditions (BCs), i.e. as boundary value problems (BVPs) [1, 2]. Under appropriate conditions on the domain and the source term, BVPs are known to be well-posed on a continuous level [3]. Amongst other physical problems, acoustic wave behavior is often described by Helmholtz equations [1], whose underlined operator is coercive [2, Section 3.6]--of the form elliptic+compact operator. Traditional schemes for solving BVPs include finite element methods (FEM) [2, 3], spectral methods or boundary element methods (BEM) [4, 5], the latter being commonly used for unbounded domains. These techniques benefit from an enriched theory, including precise convergence bounds for both the solution error and iterative solvers [2]. They have been the state-of-the art solution in engineering applications over the past decades.