Taper-based scattering formulation of the Helmholtz equation to improve
  the training process of Physics-Informed Neural Networks
W. Dörfler M. Elasmi T. Laufer
This work addresses the scattering problem of an incident wave at a junction
connecting two semi-infinite waveguides, which we intend to solve using
Physics-Informed Neural Networks (PINNs). As with other deep learning-based
approaches, PINNs are known to suffer from a spectral bias and from the
hyperbolic nature of the Helmholtz equation. This makes the training process
challenging, especially for higher wave numbers. We show an example where these
limitations are present. In order to improve the learning capability of our
model, we suggest an equivalent formulation of the Helmholtz Boundary Value
Problem (BVP) that is based on splitting the total wave into a tapered
continuation of the incoming wave and a remaining scattered wave. This allows
the introduction of an inhomogeneity in the BVP, leveraging the information
transmitted during back-propagation, thus, enhancing and accelerating the
training process of our PINN model. The presented numerical illustrations are
in accordance with the expected behavior, paving the way to a possible
alternative approach to predicting scattering problems using PINNs.