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Manifold Diffusion Fields
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Proposes Manifold Diffusion Fields (MDF), a diffusion model over functions on Riemannian manifolds that uses Laplace-Beltrami eigenfunctions as an intrinsic positional coordinate system, evaluated on fixed-manifold function datasets, ERA5, and GEOM-QM9 conformers.
Problem
Learning generative models of continuous functions (fields) defined on general non-Euclidean manifolds, such as climate fields on the sphere, PDE solutions on surfaces, and molecular conformers, where ambient-space coordinates are not intrinsic.
Approach
Represent points on a manifold with the first k eigenfunctions of the Laplace-Beltrami operator as a Fourier-like intrinsic positional embedding, and formulate a DDPM-style denoising diffusion over explicit context/query coordinate-signal sets, so the model is invariant to rigid and isometric transformations of the manifold.
Key results
- On GEOM-QM9 recall, MDF reaches mean coverage 95.30 and mean AMR 0.124 versus 92.80 and 0.178 for Torsional Diffusion.
- MDF outperforms DPF on fixed manifolds, with higher COV and lower MMD, with the gap largest on high mean-curvature manifolds.
- MDF's performance is stable under rigid and isometric transformations of the training manifold, whereas DPF degrades sharply.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use MDF instead of Torsional Diffusion | GEOM-QM9 conformer generation · GEOM-QM9 | Recall Coverage (mean): 92.80 -> 95.30 | Helped |
| use MDF instead of Torsional Diffusion | GEOM-QM9 conformer generation · GEOM-QM9 | Recall AMR (mean): 0.178 -> 0.124 | Helped |
| use MDF instead of Torsional Diffusion | GEOM-QM9 conformer generation · GEOM-QM9 | Precision AMR (mean): 0.221 -> 0.169 | Helped |
| use MDF instead of Torsional Diffusion | GEOM-QM9 conformer generation · GEOM-QM9 | Precision Coverage (mean): 92.70 -> 91.50 | Hurt |
| use k=16 eigen-functions instead of the default (ablation) | MDF · GEOM-QM9 | Recall AMR (mean): 0.124 -> 0.139 | Hurt |
| use k=8 eigen-functions instead of the default (ablation) | MDF · GEOM-QM9 | Recall AMR (mean): 0.124 -> 0.162 | Hurt |
| use k=4 eigen-functions instead of the default (ablation) | MDF · GEOM-QM9 | Recall AMR (mean): 0.124 -> 0.145 | Hurt |
| use k=2 eigen-functions instead of the default (ablation) | MDF · GEOM-QM9 | Recall AMR (mean): 0.124 -> 0.152 | Hurt |
Surprising findings
- With only k=2 Laplacian eigenfunctions, MDF still generates accurate conformers on GEOM-QM9 (AMR 0.152 recall).
- On QM9 precision, Torsional Diffusion has slightly higher coverage (92.70 vs 91.50 for MDF).
Limitations
- Sampling requires iterating over all diffusion timesteps, slower than GAN-based generators.
- Transformer score network cost is high; PerceiverIO is used to mitigate it.
- Eigenfunctions are only defined on manifold points, and eigenvectors of discrete Laplacians are unique only up to sign, which can complicate transfer across discretizations.
Experimental setup
Datasets: GMM on manifolds MNIST CelebA-HQ ERA5 GEOM-QM9 Stanford bunny TOSCA human cat mesh
Benchmarks: GEOM-QM9 ERA5 CelebA-HQ on bunny GMM on bunny/human/wave
Metrics: COV MMD AMR Coverage
Baselines: DPF GASP CGCF GeoDiff GeoMol Torsional Diffusion
Builds on: DDPM Diffusion Probabilistic Fields PerceiverIO Torsional Diffusion
Closest papers by meaning
- Efficient Diffusion Models for Symmetric Manifolds (ICML 2025 · similarity 0.90)
- Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes (ICML 2024 · similarity 0.90)
- Improving the Euclidean Diffusion Generation of Manifold Data by Mitigating Score Function Singularity (NeurIPS 2025 · similarity 0.90)
- Learning Distributions on Manifolds with Free-Form Flows (NeurIPS 2024 · similarity 0.89)
- Toward a Unified Geometry Understanding : Riemannian Diffusion Framework for Graph Generation and Prediction (NeurIPS 2025 · similarity 0.88)
- Diffusion Model's Generalization Can Be Characterized by Inductive Biases toward a Data-Dependent Ridge Manifold (NeurIPS 2026 · similarity 0.88)
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