利用上同调框架等变性学习物理场的离散黎曼度量

中文摘要 AI 辅助

网格上的物理场需要区分拓扑与几何：守恒律是拓扑性的，必须精确；而几何、材料响应及各向异性耦合需从数据中学习。现有神经代理模型常通过无约束消息传递混淆这些角色。本文提出黎曼霍奇消息传递（RHMP），将这种区分转化为架构原则。RHMP固定由定向关联确定的胞腔上边缘算子（$d_k$），并学习依赖几何的传播所需的对称正定上同调度量（$H_k$）。将$H_k$视为学习得到的度量，催生了上同调框架等变性：物理传播应对隐藏上同调特征基的正交变换保持不变。RHMP通过度量加权霍奇块（$d_k^\top H_{k+1}d_k$）实现该原则，得到精确的上同调复形恒等式（$d_{k+1}d_k=0$）、非负霍奇能量、半正定算子及精确的阿贝尔曲率不变性。在涵盖流体、电磁学、规范场及变网格计算流体动力学的7个物理基准测试中，RHMP取得了最佳整体性能，且在拓扑、学习到的几何与场结构相互作用时增益最大。

英文摘要

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.

发表机构

• Princeton University（普林斯顿大学）

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