For many applications, scientific machine learning techniques are still limited in their ability to guarantee structure preservation inherent to various physical systems, and in their ability to achieve theoretic convergence rates. To address this limitation, we introduce a scientific machine learning framework based upon a partition of unity architecture; this architecture identifies physically-relevant control volumes, encoding generalized fluxes between subdomains via Whitney forms. Subsequently, this architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. The resulting differentiable parameterization of geometry may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics and while matching expected convergence rates. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in two dimensions for both linear and nonlinear physical systems. These examples highlight the convergence rates, structure preservation properties, and model reduction capabilities of the learned finite element exterior calculus architecture. In particular, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations; our approach reduces the 5.89M finite element simulation to 136 elements yet still reproduces pressure with under 0.1% error and preserves conservation.