Solving The variable-coefficient Helmholtz Using An Artificial Neural Network Approach

Abstract

We investigate a mesh-free artificial neural network (ANN) solver for the variable-coefficient Helmholtz equation, a core model for time-harmonic wave phenomena in heterogeneous media. The proposed formulation represents the solution as a smooth neural function and trains it by minimizing a composite loss that enforces the PDE residual via automatic differentiation together with Dirichlet boundary data. We evaluate the method on a controlled two-dimensional (2-D) benchmark with spatially varying wavenumber, quantify accuracy using global error metrics (L2, MAE, RMSE), and analyze accuracy cost drivers such as collocation budgeting and boundary sampling. Results indicate that the ANN approach attains competitive accuracy relative to classical discretization pipelines while avoiding explicit meshing and offering flexibility for complex geometries. We discuss practical design choices (activation, loss weighting, sampling) that are important for oscillatory solutions and heterogeneous coefficients. The present study focuses on a specific 2-D variable-coefficient case with a known target function selected for reproducibility and ablations; therefore, the findings should not be interpreted as fully general. We outline extensions to higher-frequency regimes, non-smooth coefficients, mixed boundary conditions, and three-dimensional settings, where the same formulation applies with augmented inputs and scaled collocation/memory budgets. Overall, the study provides a reproducible benchmark and guidance for deploying ANN-based solvers on variable-coefficient Helmholtz problems.