Solving ill-posed Helmholtz problems with physics-informed neural networks

Mihai Nechita

doi:10.33993/jnaat521-1305

Authors

Mihai Nechita Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy & Babeș-Bolyai University, Cluj-Napoca, Romania https://orcid.org/0000-0002-6448-912X

DOI:

https://doi.org/10.33993/jnaat521-1305

Keywords:

ill-posed problems, inverse problems, unique continuation, data assimilation, Helmholtz equation, physics-informed neural networks

Abstract views: 376

Abstract

We consider the unique continuation (data assimilation) problem for the Helmholtz equation and study its numerical approximation based on physics-informed neural networks (PINNs). Exploiting the conditional stability of the problem, we first give a bound on the generalization error of PINNs. We then present numerical experiments in 2d for different frequencies and for geometric configurations with different stability bounds for the continuation problem. The results show that vanilla PINNs provide good approximations even for noisy data in configurations with robust stability (both low and moderate frequencies), but may struggle otherwise. This indicates that more sophisticated techniques are needed to obtain PINNs that are frequency-robust for inverse problems subject to the Helmholtz equation.

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References

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