Solving The variable-coefficient Helmholtz Using An Artificial Neural Network Approach
Bibliographic record
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.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".