Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivariance
Introduces Riemannian Hodge Message Passing (RHMP), an architecture that explicitly separates topological conservation laws from learned geometric properties in physical field simulations on meshes Fixes cellular coboundaries (d_k) from oriented incidence while learning symmetric positive-definite cochain metrics (H_k) for geometry-dependent propagation Enforces cochain-frame equivariance, ensuring physical propagation is invariant to orthogonal changes of the hidden cochain feature basis Delive
Analysis
TL;DR
- Introduces Riemannian Hodge Message Passing (RHMP), an architecture that explicitly separates topological conservation laws from learned geometric properties in physical field simulations on meshes
- Fixes cellular coboundaries (d_k) from oriented incidence while learning symmetric positive-definite cochain metrics (H_k) for geometry-dependent propagation
- Enforces cochain-frame equivariance, ensuring physical propagation is invariant to orthogonal changes of the hidden cochain feature basis
- Delivers exact cochain-complex identities (d_{k+1}d_k=0), nonnegative Hodge energies, positive-semidefinite operators, and exact Abelian curvature invariance
- Achieves best overall performance across seven physical benchmarks spanning fluids, electromagnetism, gauge fields, and variable-mesh CFD
Why It Matters
This work addresses a fundamental challenge in physics-informed machine learning: the conflation of topological constraints with geometric learning in neural surrogates for physical systems. By making the topology-geometry separation an architectural principle rather than an afterthought, RHMP offers a principled path toward more reliable and physically consistent learned simulators, which is critical for applications in computational fluid dynamics, electromagnetics, and gauge field theory where conservation laws must be preserved exactly.
Technical Details
- Core Architecture: RHMP fixes cellular coboundary operators (d_k) determined by oriented incidence relations on meshes, while learning symmetric positive-definite cochain metrics (H_k) that encode geometry, material response, and anisotropic coupling from data
- Cochain-Frame Equivariance: The key inductive bias — physical propagation must remain invariant under orthogonal transformations of the hidden cochain feature basis, ensuring the learned metrics respect the underlying geometric structure
- Metric-Weighted Hodge Blocks: The propagation mechanism uses operators of the form d_k^⊤ H_{k+1} d_k, which naturally yield exact cochain-complex identities, nonnegative Hodge energies, and positive-semidefinite operators
- Theoretical Guarantees: The architecture provably maintains exact Abelian curvature invariance and the fundamental identity d_{k+1}d_k = 0, properties that unconstrained message-passing networks cannot guarantee
- Benchmarks: Evaluated across seven physical simulation benchmarks including fluid dynamics, electromagnetism, gauge fields, and variable-mesh CFD, with largest performance gains observed in scenarios where topology, learned geometry, and field structure interact complexly
Industry Insight
- The explicit separation of topology and geometry in neural architecture design should become a standard principle for physics-informed ML, particularly as the industry moves toward learned surrogates for CFD and multiphysics simulation where conservation violations can lead to catastrophic failures
- Cochain-frame equivariance offers a generalizable inductive bias that could be adapted beyond differential forms to other geometric deep learning applications involving structured data on manifolds and complexes
- The demonstrated gains in variable-mesh CFD suggest this approach could accelerate the adoption of neural surrogates in industrial simulation pipelines, where mesh adaptivity is essential but conservation law preservation remains a critical bottleneck
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