Physics-constrained surrogate models for astrophysical S -factor reconstruction and low-energy extrapolation

Abstract Astrophysical S -factor analyses require both accurate reconstruction within the measured energy domain and physically controlled behaviour when the model is used outside that domain. We extend the physics-constrained surrogate model (PCSM) framework into a two-stage strategy and apply it to 14 N ( p , γ ) 15 O and 12 C ( α , γ ) 16 O . First, an in-domain PCSM benchmark compares data-only neural surrogates with first-derivative (D1) and first-plus-second-derivative (D1+D2) hinge-loss constraints over the measured data region only. This benchmark tests whether derivative information improves the learned surrogate independently of any extrapolation task. For 14 N ( p , γ ) 15 O , the D1-PCSM model reduces the all-data MAE from 0.100 to 0.057 keV b and increases R all 2 from 0.691 to 0.765 relative to the data-only surrogate. For 12 C ( α , γ ) 16 O , the D1 and D1+D2 models preserve data fidelity while providing a controlled test of derivative and curvature consistency in a more difficult resonance-structured data set. Second, we introduce a logarithmic PCSM (log-PCSM) formulation for low-energy extrapolation, in which the network learns g ( E ) = log ⁡ S ( E ) and hence enforces S ( E ) > 0 by construction. A gap-curvature smoothness penalty is used in the extrapolation gap instead of relying on a noisy second-derivative sign constraint outside the measured region. For