mlsanityData to 14 Oct 2026

DiffusionPDE: Generative PDE-Solving Under Partial Observation

arXiv 2406.17763 ↗First posted 25 Jun 2024cs.LG cs.AI cs.CVcited by 108

On the map: Diffusion models → Flow matching, time series and science

Abstract (arXiv)

We introduce a general framework for solving partial differential equations (PDEs) using generative diffusion models. In particular, we focus on the scenarios where we do not have the full knowledge of the scene necessary to apply classical solvers. Most existing forward or inverse PDE approaches perform poorly when the observations on the data or the underlying coefficients are incomplete, which is a common assumption for real-world measurements. In this work, we propose DiffusionPDE that can simultaneously fill in the missing information and solve a PDE by modeling the joint distribution of the solution and coefficient spaces. We show that the learned generative priors lead to a versatile framework for accurately solving a wide range of PDEs under partial observation, significantly outperforming the state-of-the-art methods for both forward and inverse directions.

Key facts. Cited by 108 papers in the map (data through 14 Oct 2026). Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. Builds on 42 of its 60 references that are in the map. First posted on arXiv 25 Jun 2024.

Builds on

42 of 60 references.

Closest ancestors · 4

Same direction, newest first

  1. 2023CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems16 Dec
  2. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers10 Aug
  3. 2023A Denoising Diffusion Model for Fluid Field Prediction27 Jan
  4. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction26 Nov

Other references · 38

  1. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  2. 2020Denoising Diffusion Implicit Modelscited by 9,781
  3. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  4. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,255
  5. 2022Classifier-Free Diffusion Guidancecited by 5,847
  6. 2022Elucidating the Design Space of Diffusion-Based Generative Modelscited by 3,045
  7. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  8. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  9. 2022Diffusion Posterior Sampling for General Noisy Inverse Problemscited by 1,416
  10. 2022Diffusion Models: A Comprehensive Survey of Methods and Applicationscited by 1,114
  11. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  12. 2022Improving Diffusion Models for Inverse Problems using Manifold Constraintscited by 508
26 more
  1. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  2. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 373
  3. 2023Geometry-Informed Neural Operator for Large-Scale 3D PDEscited by 303
  4. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 297
  5. 2022Transformer for Partial Differential Equations' Operator Learningcited by 283
  6. 2019DiffTaichi: Differentiable Programming for Physical Simulationcited by 258
  7. 2023Spherical Fourier Neural Operators: Learning Stable Dynamics on the Spherecited by 233
  8. 2023State of the Art on Diffusion Models for Visual Computingcited by 127
  9. 2023Inversion-Free Image Editing with Natural Languagecited by 106
  10. 2021Real-time Neural Radiance Caching for Path Tracingcited by 91
  11. 2023Operator Learning with Neural Fields: Tackling PDEs on General Geometriescited by 84
  12. 2019Physics-Informed Neural Networks for Power Systemscited by 80
  13. 2021Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equationscited by 68
  14. 2023Neural Inverse Operators for Solving PDE Inverse Problemscited by 65
  15. 2022CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representationscited by 58
  16. 2020Learning continuous-time PDEs from sparse data with graph neural networkscited by 53
  17. 2023From Zero to Turbulence: Generative Modeling for 3D Flow Simulationcited by 49
  18. 2022Learning to Solve PDE-constrained Inverse Problems with Graph Networkscited by 37
  19. 2020Solving inverse-PDE problems with physics-aware neural networkscited by 27
  20. 2024Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problemscited by 19
  21. 2022Solving Seismic Wave Equations on Variable Velocity Models with Fourier Neural Operatorcited by 18
  22. 2019Deep learning architectures for nonlinear operator functions and nonlinear inverse problemscited by 17
  23. 2023Fourier Neural Operator Surrogate Model to Predict 3D Seismic Waves Propagationcited by 11
  24. 2024Solving Poisson Equations using Neural Walk-on-Spherescited by 11
  25. 2015A penalty method for PDE-constrained optimization in inverse problemscited by 9
  26. 2023Accelerating Particle and Fluid Simulations with Differentiable Graph Networks for Solving Forward and Inverse Problemscited by 6

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 6; 2025: 41; 2026: 61.

108 papers.

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  1. 2025Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraintscited by 52
  2. 2024Gradient-Free Generation for Hard-Constrained Systemscited by 38
  3. 2024Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modelingcited by 36
  4. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEscited by 36
  5. 2024Wavelet Diffusion Neural Operatorcited by 29
  6. 2025Generative Latent Neural PDE Solver using Flow Matchingcited by 28
  7. 2025Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraintscited by 28
  8. 2024Text2PDE: Latent Diffusion Models for Accessible Physics Simulationcited by 25
  9. 2025Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networkscited by 25
  10. 2025FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modelingcited by 20
  11. 2025Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulationcited by 19
  12. 2024Zebra: In-Context Generative Pretraining for Solving Parametric PDEscited by 16
38 more
  1. 2024Spatially-Aware Diffusion Models with Cross-Attention for Global Field Reconstruction with Sparse Observationscited by 13
  2. 2025PhySense: Sensor Placement Optimization for Accurate Physics Sensingcited by 11
  3. 2025Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generationcited by 11
  4. 2025CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operatorscited by 11
  5. 2025Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problemscited by 10
  6. 2026Decoupled Diffusion Sampling for Inverse Problems on Function Spacescited by 10
  7. 2025Generating Full-field Evolution of Physical Dynamics from Irregular Sparse Observationscited by 9
  8. 2025FourierFlow: Frequency-aware Flow Matching for Generative Turbulence Modelingcited by 8
  9. 2025VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Modelscited by 8
  10. 2025Learning Turbulent Flows with Generative Models: Super-resolution, Forecasting, and Sparse Flow Reconstructioncited by 8
  11. 2025Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generationcited by 7
  12. 2025Strictly Constrained Generative Modeling via Split Augmented Langevin Samplingcited by 6
  13. 2025LaDCast: A Latent Diffusion Model for Medium-Range Ensemble Weather Forecastingcited by 6
  14. 2025Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspectivecited by 6
  15. 2025Foundation Models for Scientific Discovery: From Paradigm Enhancement to Paradigm Transitioncited by 6
  16. 2025FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulationcited by 5
  17. 2025Predicting partially observable dynamical systems via diffusion models with a multiscale inference schemecited by 5
  18. 2026Physics-informed diffusion models in spectral spacecited by 5
  19. 2026DiffusionRollout: Uncertainty-Aware Rollout Planning in Long-Horizon PDE Solvingcited by 5
  20. 2025Multiphysics Bench: Benchmarking and Investigating Scientific Machine Learning for Multiphysics PDEscited by 4
  21. 2025Variational Digital Twinscited by 4
  22. 2025Flow marching for a generative PDE foundation modelcited by 4
  23. 2025Enforcing governing equation constraints in neural PDE solvers via training-free projectionscited by 4
  24. 2026Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computingcited by 4
  25. 2025EquiReg: Equivariance Regularized Diffusion for Inverse Problemscited by 3
  26. 2025Incomplete Data, Complete Dynamics: A Diffusion Approachcited by 3
  27. 2025PRISMA: Improving the Accuracy-Latency Frontier of Diffusion-based PDE Solvers Using Physics-Informed Spectral Attentioncited by 3
  28. 2026Latent Generative Solvers for Generalizable Long-Term Physics Simulationcited by 3
  29. 2026Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systemscited by 3
  30. 2025Autoregressive regularized score-based diffusion models for multi-scenarios fluid flow predictioncited by 2
  31. 2025PIRF: Physics-Informed Reward Fine-Tuning for Diffusion Modelscited by 2
  32. 2025Reframing Generative Models for Physical Systems using Stochastic Interpolantscited by 2
  33. 2025Efficient Generative Transformer Operators For Million-Point PDEscited by 2
  34. 2026ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidancecited by 2
  35. 2026Ultra Fast PDE Solving via Physics Guided Few-step Diffusioncited by 2
  36. 2026Stabilizing Physics-Informed Consistency Models via Structure-Preserving Trainingcited by 2
  37. 2026Function-Space Decoupled Diffusion for Forward and Inverse Modeling in Carbon Capture and Storagecited by 2
  38. 2026Ambient Physics: Training Neural PDE Solvers with Partial Observationscited by 2

50 most cited of 108 All 108 (JSON)

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  1. 2024Diffusion models as probabilistic neural operators for recovering unobserved states of dynamical systems
  2. 2026Particle-Guided Diffusion Models for Partial Differential Equations
  3. 2024ODE-DPS: ODE-based Diffusion Posterior Sampling for Inverse Problems in Partial Differential Equation
  4. 2026A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models
  5. 2026The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems
  6. 2025Unifying and extending Diffusion Models through PDEs for solving Inverse Problems
  7. 2024Physics-Informed Diffusion Models
  8. 2025Generative diffusion models from a PDE perspective