On the metastability of learning algorithms in physics-informed neural networks: a case study on Schr\"{o}dinger operators

In this manuscript, we discuss an interesting phenomenon that happens in the training of physics-informed neural networks: PINNs seem to go through metastable states during the optimization process. This behaviour is present in several dynamical systems of interest to physics and was first noticed in the Fermi-Pasta-Ulam-Tsingou model, in which the system spends a lot of time in an intermediate state, before, eventually, reaching thermalization. We concentrate on some examples of Schr\"{o}dinger equations in spatial dimension $n=1$, including the nonlinear Schr\"{odinger} equation with quintic polynomial nonlinearity, the linear Schr\"{o}dinger equation with trapping potential, and and the linear Schr\"{o}dinger equation with asymptotically constant potential.