Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivariance
Read the original on arXiv Machine Learning →The paper introduces Riemannian Hodge Message Passing (RHMP), a neural network architecture that separates topology from geometry in physical field modeling on meshes. RHMP fixes the cellular coboundaries determined by mesh incidence and learns symmetric positive‑definite cochain metrics to guide geometry‑dependent propagation, ensuring exact cochain‑complex identities and cochain‑frame equivariance. Experiments on seven benchmarks—including fluids, electromagnetism, gauge fields, and variable‑mesh CFD—show that RHMP achieves strong performance, especially when topology, learned geometry, and field structure interact.
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