mlsanityData to 7 Oct 2026

GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains

arXiv 2605.03497 ↗First posted 5 May 2026cs.LGcited by 1

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Score-based diffusion models in infinite-dimensional function spaces provide a mathematically principled framework for modelling function-valued data, offering key advantages such as resolution invariance and the ability to handle irregular discretisations. However, practical implementations have struggled to fully realise these benefits. Existing backbones like Fourier neural operators are often biased towards regular grids and fail to generalise to complex domain topologies. We introduce an architecture, GRIFDIR, for function-space diffusion models that represents generalised graph convolutional kernels as finite element functions, allowing the score network to operate directly on unstructured meshes over domains of arbitrary shape. We demonstrate the efficacy of our network architecture through a series of unconditional and conditional sampling experiments across diverse geometries, including non-convex and multiply-connected domains. Our results show that the proposed method maintains resolution invariance and achieves high fidelity in capturing functional distributions on non-trivial geometries.

Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 45 of its 55 references that are in the map. First posted on arXiv 5 May 2026.

Builds on

45 of 55 references.

Closest ancestors · 8

Same direction, newest first

  1. 2024DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transform3 Jun
  2. 2023Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation8 Mar
  3. 2023Infinite-Dimensional Diffusion Models20 Feb
  4. 2023Score-based Diffusion Models in Function Space14 Feb
  5. 2022Scalable Diffusion Models with Transformers19 Dec
  6. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  7. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  8. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov

Other references · 37

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  2. 2017PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Spacecited by 4,384
  3. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  4. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  5. 2017FiLM: Visual Reasoning with a General Conditioning Layercited by 2,286
  6. 2020Point Transformercited by 1,462
  7. 2022Diffusion Posterior Sampling for General Noisy Inverse Problemscited by 1,375
  8. 2020Learning Mesh-Based Simulation with Graph Networkscited by 840
  9. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  10. 2019Graph U-Netscited by 489
  11. 2016Geometric deep learning on graphs and manifolds using mixture model CNNscited by 468
  12. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
25 more
  1. 2022Message Passing Neural PDE Solverscited by 368
  2. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  3. 2017Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphscited by 307
  4. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  5. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
  6. 2017SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernelscited by 160
  7. 2024Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealingcited by 157
  8. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 100
  9. 2022Predicting Physics in Mesh-reduced Space with Temporal Attentioncited by 82
  10. 2023Functional Flow Matchingcited by 63
  11. 2023Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learningcited by 57
  12. 2025Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domainscited by 56
  13. 2023Conditional score-based diffusion models for Bayesian inference in infinite dimensionscited by 37
  14. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEscited by 33
  15. 2025Reduced Order Modeling with Shallow Recurrent Decoder Networkscited by 24
  16. 2025Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networkscited by 24
  17. 2023Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domainscited by 20
  18. 2024DeepJEB: 3D Deep Learning-based Synthetic Jet Engine Bracket Datasetcited by 16
  19. 2025Mesh-Informed Neural Operator : A Transformer Generative Approachcited by 11
  20. 2025Diffusion-Based Hierarchical Graph Neural Networks for Simulating Nonlinear Solid Mechanicscited by 10
  21. 2026Decoupled Diffusion Sampling for Inverse Problems on Function Spacescited by 9
  22. 2025Geometry-Informed Neural Operator Transformercited by 8
  23. 2022QuadConv: Quadrature-Based Convolutions with Applications to Non-Uniform PDE Data Compressioncited by 6
  24. 2026Supervised Guidance Training for Infinite-Dimensional Diffusion Modelscited by 3
  25. 2026Function-Space Decoupled Diffusion for Forward and Inverse Modeling in Carbon Capture and Storagecited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 1.

1 paper.

  1. 2026Christoffel-DPS: Optimal sensor placement in diffusion posterior sampling for arbitrary distributionscited by 0

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