Riemannian Diffusion Models on General Manifolds via Physics-Informed Neural Networks

• URL: http://arxiv.org/abs/2605.31106v1
• Date: Fri, 29 May 2026 10:16:47 GMT
• Title: Riemannian Diffusion Models on General Manifolds via Physics-Informed Neural Networks
• Abstract summary: We propose a general approach that approximates the heat kernel by directly solving the manifold heat equation with a physics-informed neural network (PINN)<n>The resulting surrogate enables both forward noising (heat- Kernel sampling) and conditional-score evaluation for denoising score matching.
• Score: 15.27338478539122
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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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