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In this work, we present a general framework that unites the two primarystrategies for constructing density functional approximations (DFAs):non-empirical (NE) constraint satisfaction and semi-empirical (SE) data-drivenoptimization. The proposed method employs B-splines -- bell-shaped splinefunctions with compact support -- to construct each inhomogeneity correctionfactor (ICF). This choice offers several distinct advantages over a polynomialbasis by enabling explicit enforcement of linear and non-linear constraints aswell as ICF smoothness using Tikhonov regularization and penalized B-splines(P-splines). As proof of concept, we use this approach to construct CASE21 -- aConstrained And Smoothed semi-Empirical hybrid generalized gradientapproximation that completely satisfies all but one constraint (and partiallysatisfies the remaining one) met by the PBE0 NE-DFA and exhibits enhancedperformance across a diverse set of chemical properties. As such, we argue thatthe paradigm presented herein maintains the physical rigor and transferabilityof NE-DFAs while leveraging high-quality quantum-mechanical data to improveperformance.

CASE21: Uniting Non-Empirical and Semi-Empirical Density Functional Approximation Strategies using Constraint-Based Regularization