用几何学习与拓扑分离的框架，提升物理场建模精度
Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivarianc
- 通过共链-框架等变性设计新架构，分离拓扑与几何角色
- 在7个物理任务中表现最佳，尤其在拓扑与几何耦合强时提升显著
- 适合需要精确守恒律和数据驱动几何学习的物理模拟研究者
网格上的物理场需分离拓扑与几何：守恒律为拓扑性质应精确满足，而几何、材料响应与各向异性耦合则需从数据中学习。现有神经代理常将两者混杂于无约束的消息传递中。本文提出黎曼霍奇消息传递（RHMP），将分离原则融入架构设计：固定由定向关联决定的细胞上微分算子（d_k），学习对称正定共链度量（H_k）以表征几何依赖传播。将H_k视为可学习度量，启发共链-框架等变性——物理传播应不随隐藏共链特征基的正交变换而改变。RHMP通过度量加权霍奇块（d_k^⊤ H_{k+1} d_k）实现该原则，保证精确的共链复形恒等式（d_{k+1}d_k=0）、非负霍奇能量、半正定算子及精确阿贝尔曲率不变性。在涵盖流体、电磁学、规范场与变网格CFD的七个物理基准测试中，RHMP整体性能最优，尤其在拓扑、学习几何与场结构相互作用时提升最大。
原文摘要 · Abstract (English)
Physical fields on meshes require a separation between topology and geometry: conservation laws are topological and should be exact, while geometry, material response, and anisotropic coupling must be 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. Treating $H_k$ as the learned metric motivates cochain-frame equivariance: physical propagation should be invariant to orthogonal changes of the hidden cochain feature basis. 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 achieves the best overall performance, with the largest gains when topology, learned geometry, and field structure interact.
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