RHMP framework tops 7 physics benchmarks with exact conservation

Physics simulations on meshes have long struggled with a fundamental tension: conservation laws are topological and must be exact, while geometry, material response, and anisotropic coupling must be learned from data. Existing neural surrogates typically mix these roles inside unconstrained message passing, leading to models that violate exact conservation or fail to generalize across mesh resolutions. In a new arXiv paper (2608.14556), researchers Dongzhe Zheng and Christine Allen-Blanchette propose Riemannian Hodge Message Passing (RHMP), an architecture that makes this separation an explicit design principle.

RHMP fixes the cellular coboundaries (d_k) determined by oriented incidence, while learning symmetric positive-definite cochain metrics (H_k) for geometry-dependent propagation. Treating H_k as the learned metric leads naturally to cochain-frame equivariance—physical propagation remains invariant to orthogonal changes of the hidden feature basis. The method implements this with metric-weighted Hodge blocks (d_k^T H_{k+1} d_k), which guarantee exact cochain-complex identities (d_{k+1}d_k=0), nonnegative Hodge energies, positive-semidefinite operators, and exact Abelian curvature invariance. Across seven benchmarks covering fluids, electromagnetism, gauge fields, and variable-mesh CFD, RHMP achieves the best overall performance, with the largest gains when topology, learned geometry, and field structure interact.