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

W. Dörfler, M. Elasmi, T. Laufer

TL;DR

The paper tackles scattering problems for the Helmholtz equation in waveguide junctions using Physics-Informed Neural Networks (PINNs), addressing challenges from spectral bias and the hyperbolic nature of the PDE. It introduces a taper-based scattering formulation by splitting the total field as $u = u^{\mathrm{sct}} + \chi\,u^{\mathrm{inc}}$, which injects an inhomogeneity into the BVP and leads to improved training behavior via back-propagation information. Numerical experiments show that this approach extends accurate predictions to higher wave numbers (up to at least $k=16$ in the tested setup) and accelerates convergence compared to the classical formulation, while relying on the Dirichlet-to-Neumann operator $\Lambda$ at the interfaces. The work suggests that taper-based reformulations can serve as a practical, extendable tool to enhance PINN-based predictions for scattering problems and potentially generalize to higher modes and more complex geometries.

Abstract

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.

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

TL;DR

The paper tackles scattering problems for the Helmholtz equation in waveguide junctions using Physics-Informed Neural Networks (PINNs), addressing challenges from spectral bias and the hyperbolic nature of the PDE. It introduces a taper-based scattering formulation by splitting the total field as , which injects an inhomogeneity into the BVP and leads to improved training behavior via back-propagation information. Numerical experiments show that this approach extends accurate predictions to higher wave numbers (up to at least in the tested setup) and accelerates convergence compared to the classical formulation, while relying on the Dirichlet-to-Neumann operator at the interfaces. The work suggests that taper-based reformulations can serve as a practical, extendable tool to enhance PINN-based predictions for scattering problems and potentially generalize to higher modes and more complex geometries.

Abstract

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.
Paper Structure (5 sections, 13 equations, 4 figures, 1 table)

This paper contains 5 sections, 13 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The scattering problem
  3. Physics-Informed Neural Networks
  4. Numerical Studies
  5. Summary and Conclusion

Figures (4)

  • Figure 1: Illustration of the considered type of waveguide junctions $\Omega$. Thereby, $\Omega_+$ and $\Omega_-$ denote two semi-infinite waveguides connected by $\Omega$ at the interfaces $\Gamma_-$ (at $x=-b$) and $\Gamma_+$ (at $x=b$), respectively. $\Gamma_{0,1}$ and $\Gamma_{0,2}$ designate the remaining boundaries of $\Omega$ott2015domain.
  • Figure 2: Visualization of the real part of the PINN's prediction $\hat{u}:= u_\theta$ using \ref{['prob:helmholtzBVP']} after $50000$Adam training steps.
  • Figure 3: Visualization of the real part of the reconstructed solution $\hat{u}:=u_\theta + \chi u^\mathrm{inc}$ and its constituents using \ref{['prob:taperBasedFormulation']} after $50000$Adam training steps.
  • Figure 4: Comparison of the learning dynamics using the relative error \ref{['eq:relError']} w.r.t. the real part of the predicted solution $\hat{u}$ (denoted by $\Re\{\hat{u}\}$) from Problems \ref{['prob:helmholtzBVP']} and \ref{['prob:taperBasedFormulation']} with $k=8$. The relative error is evaluated for visualization purpose once each $100$ iterations.

Theorems & Definitions (1)

  • Definition 2.1: Dirichlet-to-Neumann (DtN) operator