An activation-selection neural network produced compact mathematical formulas for quantum energy levels in three-site and four-site spin-chain benchmarks, but the study did not show that the approach predicts those levels better than fixed-basis quadratic least squares. Where researchers supplied a suitable quadratic basis in advance, least-squares fits matched or slightly outperformed the neural network.
The search for a usable formula
The preprint tests whether learned activation selection can produce compact analytical approximations to parameter-dependent Hamiltonian eigenvalues beyond direct perturbative and least-squares approaches. In plain terms, Hamiltonian eigenvalues are the energy levels being approximated as the model's parameters change.
The network uses dimensionless ratios for parameters and energy, and its library contains 0, x and x squared. Stacked layers let it generate polynomial expressions of progressively higher degree. The design allows training to select and compose candidate analytical functions before extracting an explicit formula.
After training, the learned coefficients and linear parameters were composed and simplified symbolically. Coefficients below 10^-3 in absolute value were removed, and the remainder were rounded to three decimal places.
The primary three-site test used 100,000 parameter pairs generated with NumPy seed 42. The primary four-site test used 10,000 examples obtained by linear interpolation from a 200 by 200 grid. Both primary fits used random 80% training and 20% test splits with seed 42.
The first test exposed a trade-off
In the three-site benchmark, overall mean squared error, or MSE, was 3.20 x 10^-4, while root mean squared error, or RMSE, was 1.79 x 10^-2. The branchwise R2 scores, a measure of fit quality, were 0.9994, 0.9888, 0.9483 and 0.9979 for epsilon 1 through epsilon 4. The network therefore fit the energy surfaces well overall, but not every branch equally well. These figures were calculated before coefficient thresholding, symmetry averaging and rounding.
The harder comparison came near a degeneracy, where the leading perturbative splitting in the nonuniform three-site chain is radial and nonpolynomial. The polynomial library can approximate that structure, but cannot represent it exactly at the degeneracy. Across the complete sample, symmetry-constrained quadratic least squares had an RMSE of 1.62 x 10^-2, compared with 1.79 x 10^-2 for the unrounded ASN. Near the degeneracy, a least-squares model using radial features had local RMSE 6.58 x 10^-5, compared with 1.74 x 10^-3 for the quadratic model and 4.28 x 10^-5 for the perturbative reference. On held-out data, the radial-feature model's RMSE was 1.40 x 10^-2.
The larger chain held up better
In the four-site test, the ASN reproduced all eight sampled branches across 0 <= r <= 3. On held-out data, MSE was 6.2 x 10^-5 and RMS deviation was 7.9 x 10^-3; branchwise R2 ranged from 0.9918 to 0.9998. Once extracted, the branch formulas were quadratic polynomials with no more than three terms. The largest deviations occurred near the upper edge of the sampled range, and the paper does not establish how the expressions behave beyond that domain.
An independently written four-site implementation produced a similar result. Its held-out MSE was 5.67 x 10^-5 and RMSE was 7.53 x 10^-3, with R2 scores from 0.9935 to 0.9998. All 11 activation-selection coefficient vectors had support size one, meaning one selected activation remained in each vector. This was a separate training and evaluation run, not the identical fitted model. On that same sample and split, quadratic-basis least squares was slightly more accurate, with MSE 4.93 x 10^-5 and RMSE 7.02 x 10^-3.
What the results do and do not claim
The authors frame the ASN's value as structural rather than predictive. It can learn to select and compose candidate analytical functions and then extract a compact global approximation. But when the appropriate fixed basis is already known, the results do not show an intrinsic accuracy advantage over quadratic least squares.
These conclusions apply to the reported three-site and four-site benchmarks and their sampled parameter domains. The paper gives no guarantee that the extracted expressions will extrapolate beyond those domains and calls for further work on consistent branch assignment, multiple training seeds and exact spectral constraints.
The manuscript front matter is dated 25 August 2026. The authors state that the supporting code and datasets are available in the NMRKan repository.
Paper data and sources
Original title: Interpretable Activation-Selection Neural Networks for Symbolic Regression of Parameter-Dependent Hamiltonian Eigenvalues
Authors: Alexander Yu. Kharin, Kirill F. Sheberstov
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text