  Physics-informed neural networks (PINNs) were recently proposed in [1] as an
alternative way to solve partial differential equations (PDEs). A neural
network (NN) represents the solution while a PDE-induced NN is coupled to the
solution NN, and all differential operators are treated using automatic
differentiation. Here, we first employ the standard PINN and a stochastic
version, sPINN, to solve forward and inverse problems governed by a nonlinear
advection-diffusion-reaction (ADR) equation, assuming we have some sparse
measurements of the concentration field at random or pre-selected locations.
Subsequently, we attempt to optimize the hyper-parameters of sPINN by using the
Bayesian optimization method (meta-learning), and compare the results with the
empirically selected hyper-parameters of sPINN.
