Nonintrusive reduced basis approximation to the solution of the Helmholtz equation: The magnetotellurics case
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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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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it