mlsanityData to 14 Oct 2026

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

arXiv 2605.31106 ↗First posted 29 May 2026cs.LGcited by 0

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

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 S2, SO(3), SPD(n), and permutation-quotiented point clouds.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 26 of its 50 references that are in the map. First posted on arXiv 29 May 2026.

Builds on

26 of 50 references.

Closest ancestors · 8

Same direction, newest first

  1. 2025Efficient Diffusion Models for Symmetric Manifolds27 May
  2. 2023SPD-DDPM: Denoising Diffusion Probabilistic Models in the Symmetric Positive Definite Space13 Dec
  3. 2022Riemannian Diffusion Models16 Aug
  4. 2022Riemannian Diffusion Schr"odinger Bridge7 Jul
  5. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  6. 2022Riemannian Score-Based Generative Modelling6 Feb
  7. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  8. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 18

  1. 2016PointNet: Deep Learning on Point Sets for 3D Classification and Segmentationcited by 5,248
  2. 2019A Comprehensive Survey on Graph Neural Networkscited by 2,919
  3. 2021E(n) Equivariant Graph Neural Networkscited by 965
  4. 2022Equivariant Diffusion for Molecule Generation in 3Dcited by 677
  5. 2018Conditional Neural Processescited by 426
  6. 2023Flow Matching on General Geometriescited by 265
  7. 2022Torsional Diffusion for Molecular Conformer Generationcited by 263
  8. 2021How Faithful is your Synthetic Data? Sample-level Metrics for Evaluating and Auditing Generative Modelscited by 196
  9. 2023SE(3)-Stochastic Flow Matching for Protein Backbone Generationcited by 150
  10. 2022SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusioncited by 127
  11. 2023An Expert's Guide to Training Physics-informed Neural Networkscited by 126
  12. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 79
6 more
  1. 2023Separable Physics-Informed Neural Networkscited by 66
  2. 2023Navigating the Design Space of Equivariant Diffusion-Based Generative Models for De Novo 3D Molecule Generationcited by 50
  3. 2024SemlaFlow -- Efficient 3D Molecular Generation with Latent Attention and Equivariant Flow Matchingcited by 41
  4. 2023SE(3) Diffusion Model-based Point Cloud Registration for Robust 6D Object Pose Estimationcited by 33
  5. 2024PPFlow: Target-aware Peptide Design with Torsional Flow Matchingcited by 26
  6. 2025Riemannian Flow Matching for Brain Connectivity Matrices via Pullback Geometrycited by 13

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  1. 2023Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes
  2. 2025Be Tangential to Manifold: Discovering Riemannian Metric for Diffusion Models
  3. 2026Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity
  4. 2023Diffusion Models for Constrained Domains
  5. 2026Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine
  6. 2025Riemannian Denoising Diffusion Probabilistic Models
  7. 2023Manifold Diffusion Fields
  8. 2026Sampling from multi-modal distributions on Riemannian manifolds with training-free stochastic interpolants