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While AI and machine learning continue to make rapid progress, the ad hoc construction of model architectures presents major challenges for developing scientific machine learning methods that preserve the theoretical guarantees underpinning conventional modeling and simulation. In this talk, we present recent work developing hybrid transformer–finite element architectures that incorporate the design principles of finite element exterior calculus (FEEC). Using this framework, we formulate equality-constrained optimization problems that allow us to reverse engineer reduced-order descriptions of physical systems from data while preserving topological structure. We provide an overview of several results based on this approach: mixed finite element methods yield autoregressive models that outperform foundation models with 1000× fewer parameters; metriplectic brackets preserve nonequilibrium statistics in coarse-grained systems; and coordinate-free representations of geometry and physics produce models that remain accurate on geometries unseen during training.