Optimal Control in Hilbert Complex Spaces with Finite Element Exterior Calculus

What the math gives to ML

The paper separates a physical field into a gauge-fixed local component and a harmonic component carrying global circulation or flux. This suggests a topology-aware neural representation for meshes and discretized differential forms: process local variations with message passing while maintaining a small explicit latent state for global modes. A period map gives a concrete interface between this latent state and measurable global quantities, avoiding the need for bounded-depth networks to reconstruct noncontractible structure through local propagation alone. The most direct test is a hybrid Hodge-neural output head on multiply connected meshes, evaluated on global-period prediction and transfer to refined meshes.