Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivariance

Abstract

Physical fields on meshes require a separation between topology and geometry: differential identities should remain exact, while geometry, material response, and anisotropic coupling are learned from data. Existing neural surrogates often mix these roles inside unconstrained message passing. We introduce Riemannian Hodge Message Passing (RHMP), which turns this separation into an architectural principle. RHMP fixes the cellular coboundaries ($d_k$) determined by oriented incidence and learns symmetric positive-definite cochain metrics ($H_k$) for geometry-dependent propagation. The learned metric defines how cochain fields are measured and coupled; its invariance to orthogonal changes of the hidden feature basis yields cochain-frame-equivariant propagation. RHMP implements this principle with metric-weighted Hodge blocks ($d_k^\top H_{k+1}d_k$), yielding exact cochain-complex identities ($d_{k+1}d_k=0$), nonnegative Hodge energies, positive-semidefinite operators, and exact Abelian curvature invariance. Across seven physical benchmarks spanning fluids, electromagnetism, gauge fields, and variable-mesh CFD, RHMP performs strongly across these systems, with the largest gains when topology, learned geometry, and field structure interact.