  In this work, we present a general framework that unites the two primary
strategies for constructing density functional approximations (DFAs):
non-empirical (NE) constraint satisfaction and semi-empirical (SE) data-driven
optimization. The proposed method employs B-splines -- bell-shaped spline
functions with compact support -- to construct each inhomogeneity correction
factor (ICF). This choice offers several distinct advantages over a polynomial
basis by enabling explicit enforcement of linear and non-linear constraints as
well as ICF smoothness using Tikhonov regularization and penalized B-splines
(P-splines). As proof of concept, we use this approach to construct CASE21 -- a
Constrained And Smoothed semi-Empirical hybrid generalized gradient
approximation that completely satisfies all but one constraint (and partially
satisfies the remaining one) met by the PBE0 NE-DFA and exhibits enhanced
performance across a diverse set of chemical properties. As such, we argue that
the paradigm presented herein maintains the physical rigor and transferability
of NE-DFAs while leveraging high-quality quantum-mechanical data to improve
performance.
