Iraklis Spyrou, Christos Tsepas, Vaia Prassa, Christoforos Rekatsinas
Abstract
The Woods–Saxon mean-field approach provides a compact description of bound single-particle motion in finite nuclei, while physics-informed neural networks offer a differentiable route for solving the associated inverse quantum problem from sparse spectral data. This study develops a probabilistic physics-informed framework in which WaveNet represents the separated single-particle wavefunction and ParamNet maps selected spectra, nuclear descriptors, and quantum numbers to a learned output distribution over six global Woods–Saxon parameters. The Hamiltonian includes the central Woods–Saxon interaction, the proton Coulomb term, and the spin–orbit contribution, while the training process enforces spectral energy consistency, Schrödinger-equation residuals, boundary conditions, normalization, orthogonality, spin–orbit splitting constraints, and latent regularization; the distribution-mean estimator serves as the primary selection-free parameter estimate and is validated with an independent finite-difference radial solver. Synthetic closure tests based on the Seminole and Wahlborn parameterizations recover all six parameters with sub-percent relative errors and reproduce the reference spectra with mean absolute deviations of 0.0109 MeV and 0.0131 MeV, respectively. For experimental spectra using the Wahlborn potential expression, the distribution-mean estimator reduces the all-state mean absolute error from 1.0783 to 0.8303 MeV. Using the Seminole expression, the distribution-mean estimator attains an all-state mean absolute error of 0.8068 MeV, close to the 0.7969 MeV obtained with the Seminole reference parameters. This comparable accuracy is achieved using 42 selected experimental levels, approximately 51% fewer spectral targets than those used in the Seminole reference calibration. The results indicate that sparse but structured single-particle spectra can constrain global Woods–Saxon interactions within a complementary differentiable framework that addresses both the forward eigenvalue problem and inverse parameter identification while supplying a learned output spread as a model-derived, qualitative measure of parameter stiffness.