  The exchange-correlation (XC) functional in density functional theory is used
to approximate multi-electron interactions. A plethora of different functionals
is available, but nearly all are based on the hierarchy of inputs commonly
referred to as "Jacob's ladder." This paper introduces an approach to construct
XC functionals with inputs from convolutions of arbitrary kernels with the
electron density, providing a route to move beyond Jacob's ladder. We derive
the variational derivative of these functionals, showing consistency with the
generalized gradient approximation (GGA), and provide equations for variational
derivatives based on multipole features from convolutional kernels. A
proof-of-concept functional, PBEq, which generalizes the PBE$\alpha$ framework
where $\alpha$ is a spatially-resolved function of the monopole of the electron
density, is presented and implemented. It allows a single functional to use
different GGAs at different spatial points in a system, while obeying PBE
constraints. Analysis of the results underlines the importance of error
cancellation and the XC potential in data-driven functional design. After
testing on small molecules, bulk metals, and surface catalysts, the results
indicate that this approach is a promising route to simultaneously optimize
multiple properties of interest.
