Riemannian Diffusion Models on General Manifolds via Physics-Informed Neural Networks
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910271980699648 |
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| author | Ko, Gyeonghoon Lee, Juho |
| author_facet | Ko, Gyeonghoon Lee, Juho |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_31106 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Riemannian Diffusion Models on General Manifolds via Physics-Informed Neural Networks Ko, Gyeonghoon Lee, Juho Machine Learning 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. |
| title | Riemannian Diffusion Models on General Manifolds via Physics-Informed Neural Networks |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2605.31106 |