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Deep learning in shallow waters: solution of shallow water equations using physics-informed neural networks

Malamas Ilias

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Summary

In this thesis we use Physics-informed Neural Networks (PINNs) to solve the Shallow Water Equations (SWE). We provide some insight into the novel idea of PINNs, which constitute a deviation from the rigorous context of the supervised learning paradigm, in the sense that no experimental or simulation data are necessary to train the neural network to solve the SWE, making them the solver of choice in cases where the production of labelled data is costly, time-consuming, or even impossible. We first provide an outline of the system of Partial Differential Equations (PDEs), that describe the SWEs and include some principal properties. We then introduce the idea of using the PINNs as an “unconventional” solver to those PDEs. In order to validate the solver, we engage the PINNs in several benchmark problems of increasing numerical difficulty, in order to prove the adequacy of the PINN idea as a SWE solver. In the sequel, we focus on the effect of the sampling strategy of the training points (domain and boundary) that are used to train the PINN, on the performance of the PINN, in an effort to shed some light on this aspect of the PINN training, when they are used to solve the SWE, applied on a demanding, Riemann problem.

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