Nonintrusive reduced basis approximation to the solution of the Helmholtz equation: The magnetotellurics case
Bibliographic record
Abstract
ABSTRACT Electromagnetic wave propagation is commonly modeled using the Helmholtz partial differential equation, which plays a significant role in geophysical studies, such as magnetotellurics forward modeling. Although analytical solutions exist for layered media, most geophysical applications depend on numerical finite-difference, finite-element, or finite-volume solvers. These traditional methods are computationally demanding, particularly for large-scale problems and workflows that require repeated evaluations, such as real-time or probabilistic inversions. Reduced basis (RB) techniques have been developed to accelerate finite-element solvers by reducing the stiffness matrix and nodal forces vector size. However, these methods rely on explicit access to the stiffness matrix, which can limit their applicability. We present a nonintrusive data-driven approach, adapted for the first time to magnetotellurics forward modeling, that eliminates the need for explicit stiffness matrix availability and is compatible with various numerical solvers. Using a predefined parameter domain, we construct a snapshot matrix from high-fidelity solutions generated for a subset of model parameters. Proper orthogonal decomposition is then applied to extract RB, and a neural network is trained to map the model space to the reduced coefficient space. This enables rapid evaluation of the Helmholtz equation, achieving a speed-up of four orders of magnitude compared with traditional solvers, with average median errors of 9% transverse magnetic (TM) mode and 2% transverse electric (TE) mode. Further accuracy improvements are achieved by incorporating a minimal set of high-fidelity observations and leveraging the fast evaluation to regularize an inverse problem, reducing errors to 2% for the TM mode and 1.5% for the TE mode. These results highlight this approach’s potential to dramatically decrease computational costs while maintaining accuracy, making it a flexible and scalable tool for efficient geophysical forward evaluations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 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 source (direct Gemma or distilled Codex), 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".