mlsanityData to 7 Oct 2026

Generative Learning of the Solution of Parametric Partial Differential Equations Using Guided Diffusion Models and Virtual Observations

arXiv 2408.00157 ↗First posted 31 Jul 2024cs.LG physics.comp-ph physics.flu-dyncited by 16

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

Abstract (arXiv)

We introduce a generative learning framework to model high-dimensional parametric systems using gradient guidance and virtual observations. We consider systems described by Partial Differential Equations (PDEs) discretized with structured or unstructured grids. The framework integrates multi-level information to generate high fidelity time sequences of the system dynamics. We demonstrate the effectiveness and versatility of our framework with two case studies in incompressible, two dimensional, low Reynolds cylinder flow on an unstructured mesh and incompressible turbulent channel flow on a structured mesh, both parameterized by the Reynolds number. Our results illustrate the framework's robustness and ability to generate accurate flow sequences across various parameter settings, significantly reducing computational costs allowing for efficient forecasting and reconstruction of flow dynamics.

Key facts. Cited by 16 papers in the map (data through 7 Oct 2026). Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. Builds on 20 of its 41 references that are in the map. First posted on arXiv 31 Jul 2024.

Builds on

20 of 41 references.

Closest ancestors · 5

Same direction, newest first

  1. 2023CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems16 Dec
  2. 2023Uncertainty-aware Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models8 Dec
  3. 2023Bayesian Conditional Diffusion Models for Versatile Spatiotemporal Turbulence Generation14 Nov
  4. 2023Debias Coarsely, Sample Conditionally: Statistical Downscaling through Optimal Transport and Probabilistic Diffusion Models24 May
  5. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction26 Nov

Other references · 15

  1. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  2. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,185
  3. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,322
  4. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  5. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  6. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  7. 2019Machine Learning for Fluid Mechanicscited by 380
  8. 2023GenCast: Diffusion-based ensemble forecasting for medium-range weathercited by 169
  9. 2018Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networkscited by 155
  10. 2020Transformers for Modeling Physical Systemscited by 105
  11. 2020Multiscale Simulations of Complex Systems by Learning their Effective Dynamicscited by 86
  12. 2022Predicting Physics in Mesh-reduced Space with Temporal Attentioncited by 82
3 more
  1. 2019Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systemscited by 25
  2. 2021Physics-aware, probabilistic model order reduction with guaranteed stabilitycited by 12
  3. 2023Interpretable learning of effective dynamics for multiscale systemscited by 11

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 4; 2025: 9; 2026: 3.

16 papers.

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  1. 2024Spatially-Aware Diffusion Models with Cross-Attention for Global Field Reconstruction with Sparse Observationscited by 13
  2. 2025MORPH: PDE Foundation Models with Arbitrary Data Modalitycited by 12
  3. 2024CoNFiLD-inlet: Synthetic Turbulence Inflow Using Generative Latent Diffusion Models with Neural Fieldscited by 8
  4. 2025PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulationscited by 8
  5. 2025LaDCast: A Latent Diffusion Model for Medium-Range Ensemble Weather Forecastingcited by 6
  6. 2025Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Networkcited by 6
  7. 2024Advancing Generalization in PINNs through Latent-Space Representationscited by 5
  8. 2025Rectified Flows for Fast Multiscale Fluid Flow Modelingcited by 4
  9. 2025Autoregressive regularized score-based diffusion models for multi-scenarios fluid flow predictioncited by 2
  10. 2025Point-wise Diffusion Models for Physical Systems with Shape Variations: Application to Spatio-temporal and Large-scale systemcited by 2
  11. 2025Generative Urban Flow Modeling: From Geometry to Airflow with Graph Diffusioncited by 2
  12. 2024Stochastic Reconstruction of Gappy Lagrangian Turbulent Signals by Conditional Diffusion Modelscited by 1
4 more
  1. 2025Enabling Probabilistic Learning on Manifolds through Double Diffusion Mapscited by 0
  2. 2026GenDA: Generative Data Assimilation on Complex Urban Areas via Classifier-Free Diffusion Guidancecited by 0
  3. 2026Generative AI for material design: A mechanics perspective from burgers to mattercited by 0
  4. 2026Variational Parameter Calibration with Physics-Aware Latent-Space Surrogatescited by 0

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  1. 2025Physics-aware generative models for turbulent fluid flows through energy-consistent stochastic interpolants
  2. 2024Unfolding Time: Generative Modeling for Turbulent Flows in 4D
  3. 2026Learning Flow Distributions via Projection-Constrained Diffusion on Manifolds
  4. 2024Generative Learning for Forecasting the Dynamics of Complex Systems
  5. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observation
  6. 2026Learning to Discretize: Diffusion-Based Adaptive Mesh with Spectral Guidance
  7. 2026Divergence-Free Diffusion Models for Incompressible Fluid Flows
  8. 2024Generative AI for fast and accurate statistical computation of fluids