Schwarz waveform relaxation-learning for advection-diffusion-reaction equations

Abstract

This paper develops a physics-informed neural network (PINN) combined with a Schwarz waveform relaxation (SWR) method for solving local and nonlocal advection-diffusion-reaction equations. Specifically, we derive the algorithm by constructing subdomain-dependent local solutions by minimizing local loss functions, allowing the decomposition of the training process in different domains in an embarrassingly parallel procedure. Provided the convergence of PINN, the overall proposed algorithm is convergent. By constructing local solutions, one can, in particular, adapt the depth of the deep neural networks, depending on the solution's spectral space and time complexity in each subdomain. One of the main advantages of using NN compared to standard solvers, is that the PINN algorithm introduces some learning in the SWR algorithm allowing for an acceleration of the overall algorithm, especially close to SWR convergence. We present some numerical experiments based on classical and Robin-SWR algorithms to illustrate the performance and comment on the convergence of the proposed method.