Physics-constrained Surrogate Modeling with Savitzky–Golay Derivative Constraints for Non-monotone Physical Systems

Data-driven surrogate models become physically inadmissible when the data are sparse. Physics-informed neural networks impose a governing differential equation as a residual, which presupposes that the equation is known. This work concerns the case in which it is not, and physical insight is qualitative. A physics-constrained surrogate modeling framework is introduced in two stages. In the first, derivative signs prescribed by transport theory for the critical slab under linearly anisotropic scattering are imposed as hinge penalties on the surrogate derivatives. In the second, the signs need not be known in advance: each is estimated point-wise with a Savitzky–Golay filter and may reverse within a variable, extending the constraint to non-monotone responses. Only the sign is constrained, never the magnitude. On the criticality data the unconstrained network satisfies the inequalities unaided and the constraint is inactive, a negative control. On the 14C cross section it reduces the mean violation threefold over 25 random splits without loss of accuracy. A single global sign makes matters worse, as does gradient boosting under hard monotonicity: both meet the constraint by suppressing the derivative. Calibration of the sign estimator proves decisive; a density scaling rule for the smoothing window is given.