New Neural Network Improves Physical Field Simulations on Meshes

Researchers have developed a new neural network architecture called Riemannian Hodge Message Passing (RHMP) to enhance the simulation of physical fields on meshes. This approach fundamentally separates the topological aspects, which govern conservation laws and should be exact, from the geometric and material properties that need to be learned from data. Unlike existing neural surrogates that often blend these roles, RHMP fixes cellular coboundaries and learns symmetric positive-definite cochain metrics to handle geometry-dependent propagation. A key innovation of RHMP is its adherence to cochain-frame equivariance, ensuring that physical propagation remains invariant to orthogonal changes in the hidden cochain feature basis. This principle is implemented through metric-weighted Hodge blocks, which guarantee exact cochain-complex identities, non-negative Hodge energies, and positive-semidefinite operators. Evaluated across seven diverse physical benchmarks, including fluids, electromagnetism, and CFD, RHMP consistently demonstrated the best overall performance, particularly excelling in scenarios where topology, learned geometry, and field structure interact complexly.