Probabilistic physics-informed neural solvers for Woods-Saxon parameter identification: A coupled forward-inverse approach

We introduce a probabilistic physics-informed learning framework that couples quantumstate reconstruction with dataset-level identification of globally shared Hamiltonian parameters from bound-state spectra. The Woods–Saxon inverse problem provides a physically interpretable test case involving multiple nuclei, neutron and proton sectors, and Coulomb and spin–orbit contributions. WaveNet represents separated radial, polar, and azimuthal wavefunction components, while ParamNet combines normalized radial probes, observed energies, nuclear descriptors, and quantum numbers to infer a distribution over six global parameters. Training jointly enforces Rayleigh-energy consistency, separated Schr ̈odinger residuals, boundary and phase conditions, explicit normalization, selective orthogonality, spin–orbit splitting consistency, and Kullback–Leibler regularization. The physics-informed formulation supports learning from limited spectral observations without supervised wavefunction labels, using only 35 experimental energies from four nuclei. Spectral evaluation uses the physical-space distribution mean. The learned standard deviations describe modeldependent parameter variability. In synthetic closure tests, all six generating parameters are recovered with sub-percent relative errors. Reintroducing the inferred means into an independent finite-difference solver gives spectral mean absolute errors of 0.0133 MeV for the Seminole parameterization and 0.0071 MeV for Wahlborn. For experimental spectra, the inferred Seminole interaction gives an aggregate mean absolute error of 0.7812 MeV, compared with 0.7969 MeV for the reference interaction and 0.8492 MeV for a finite-difference least-squares fit using the same identification levels. Identification within the Wahlborn expression reduces the corresponding error from 1.0783 to 0.9256 MeV. The framework provides a reusable and robust scientific-machine-learning formulation for inverse quantum eigenvalue problems in which globally shared parameters and physically constrained states must be identified jointly.