An improved neural network method for solving the Schrödinger equation
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Bibliographic record
Abstract
We propose a neural network (NN) based algorithm for calculating vibrational energies and wave functions and apply it to problems in 2-, 4-, and 6-dimensions. By using neurons as basis functions and methods of nonlinear optimization, we are able to compute three states of a 6-D Hamiltonian using only 50 basis functions. In a standard direct product basis, thousands of basis functions would be necessary. Previous NN methods for solving the Schrödinger equation computed one level at a time and optimized all of the parameters using expensive nonlinear optimization methods. Using our approach, linear coefficients in the NN representation of wave functions are determined with methods of linear algebra and many levels are computed at the same time from one set of nonlinear NN parameters. In addition, we use radial basis function neurons to ensure the correct boundary conditions. The use of linear algebra methods makes it possible to treat systems of higher dimensionality.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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