用物理神经网络逼近流形上的热核，实现通用流形生成建模。
Riemannian Diffusion Models on General Manifolds via Physics-Informed Neural Networks
- 用物理信息神经网络直接求解流形热方程，近似热核。
- 在球面、旋转群等多类流形上实现采样与去噪得分匹配。
- 适用于复杂流形数据生成，如点云排列商空间。
黎曼扩散模型通过流形上的随机扩散方程将基于得分的生成建模推广到流形支持的数据。然而，训练需要从流形热核采样并对其求导，而闭式解仅在少数高度对称流形上存在。本文提出一种通用方法：通过物理信息神经网络（PINN）直接求解流形热方程来逼近热核。给定显式流形定义后，选择坐标系，推导对应的热（福克-普朗克）方程及短时间渐近近似，并训练PINN学习对数热核。所得代理模型可同时实现前向加噪（热核采样）和条件得分评估，用于去噪得分匹配。方法在多种流形上验证，包括 $S^2$、$SO(3)$、$ ext{SPD}(n)$ 以及排列商点云。
原文摘要 · Abstract (English)
Riemannian diffusion models generalize score-based generative modeling to manifold-supported data via stochastic diffusion equations on the manifold. However, training requires sampling from and differentiating the manifold heat kernel, which is rarely available in closed form beyond a few highly symmetric manifolds. We propose a general approach that approximates the heat kernel by directly solving the manifold heat equation with a physics-informed neural network (PINN). Given an explicit manifold specification, we choose a coordinate system, derive the corresponding heat (Fokker--Planck) equation and a short-time asymptotic approximation, and then train a PINN to learn the log heat kernel. The resulting surrogate enables both forward noising (heat-kernel sampling) and conditional-score evaluation for denoising score matching. We demonstrate the method on diverse manifolds including $S^2$, $SO(3)$, $\mathrm{SPD}(n)$, and permutation-quotiented point clouds.
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