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

We consider physics-informed neural networks [Raissi et al., J. Comput. Phys. 278 (2019) 686-707] for forward physical problems. In order to find optimal PINNs configuration, we introduce a hyper-parameter tuning procedure via Gaussian processes-based Bayesian optimization. We apply the procedure to Helmholtz problems for bounded domains and conduct a thorough study, focusing on: (i) performance, (ii) the collocation points density $r$ and (iii) the frequency $\kappa$, confirming the applicability and necessity of the method. Numerical experiments are performed in two and three dimensions, including comparison to finite element methods.