Taper-based scattering formulation of the Helmholtz equation to improve the training process of Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) face significant challenges when applied to problems governed by the Helmholtz equation, primarily due to its indefinite structure and the well-known spectral bias of neural networks. These issues become particularly pronounced at high wavenumbers. To address this, we propose a novel stabilization strategy that improves the learning capabilities of PINNs by recasting the original boundary value problem into an equivalent one that is more amenable to training. Specifically, we reformulate the model problem by decomposing the total wave into a tapered continuation of the incoming wave and a residual scattered field. This allows for the introduction of an inhomogeneity in the boundary value problem, which improves the flow of information during backpropagation. To illustrate the issue and highlight the effectiveness of our approach, we consider a simplified scattering problem involving an incident wave at the junction of two semi-infinite waveguides, a fundamental setting in communication technology. The numerical results presented are in accordance with the expected behavior, paving the way for a possible alternative approach to predict scattering problems using PINNs.