mlsanityData to 7 Oct 2026

Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator

arXiv 2408.02965 ↗First posted 6 Aug 2024cs.LG math.DS physics.comp-phcited by 12

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

Abstract (arXiv)

Closure models are widely used in simulating complex multiscale dynamical systems such as turbulence and the earth system, for which direct numerical simulation that resolves all scales is often too expensive. For those systems without a clear scale separation, deterministic and local closure models often lack enough generalization capability, which limits their performance in many real-world applications. In this work, we propose a data-driven modeling framework for constructing stochastic and non-local closure models via conditional diffusion model and neural operator. Specifically, the Fourier neural operator is incorporated into a score-based diffusion model, which serves as a data-driven stochastic closure model for complex dynamical systems governed by partial differential equations (PDEs). We also demonstrate how accelerated sampling methods can improve the efficiency of the data-driven stochastic closure model. The results show that the proposed methodology provides a systematic approach via generative machine learning techniques to construct data-driven stochastic closure models for multiscale dynamical systems with continuous spatiotemporal fields.

Key facts. Cited by 12 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 56 of its 104 references that are in the map. First posted on arXiv 6 Aug 2024.

Builds on

56 of 104 references.

Closest ancestors · 11

Same direction, newest first

  1. 2024Pi-fusion: Physics-informed diffusion model for learning fluid dynamics6 Jun
  2. 2024Physics-Informed Diffusion Models21 Mar
  3. 2023CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems16 Dec
  4. 2023Bayesian Conditional Diffusion Models for Versatile Spatiotemporal Turbulence Generation14 Nov
  5. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers10 Aug
  6. 2023Synthetic Lagrangian Turbulence by Generative Diffusion Models17 Jul
  7. 2023DiffSTG: Probabilistic Spatio-Temporal Graph Forecasting with Denoising Diffusion Models31 Jan
  8. 2023A Denoising Diffusion Model for Fluid Field Prediction27 Jan
  9. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction26 Nov
  10. 2021CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation7 Jul
  11. 2021Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting28 Jan

Other references · 45

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  2. 2021High-Resolution Image Synthesis with Latent Diffusion Modelscited by 18,694
  3. 2020Denoising Diffusion Implicit Modelscited by 9,655
  4. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  5. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,185
  6. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  7. 2022Classifier-Free Diffusion Guidancecited by 5,717
  8. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  9. 2022Elucidating the Design Space of Diffusion-Based Generative Modelscited by 2,994
  10. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  11. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,170
  12. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
33 more
  1. 2023Consistency Modelscited by 1,778
  2. 2021Palette: Image-to-Image Diffusion Modelscited by 1,296
  3. 2022On Distillation of Guided Diffusion Modelscited by 721
  4. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  5. 2019Sliced Score Matching: A Scalable Approach to Density and Score Estimationcited by 381
  6. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
  7. 2021Factorized Fourier Neural Operatorscited by 237
  8. 2016A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluationcited by 236
  9. 2021Gotta Go Fast When Generating Data with Score-Based Modelscited by 218
  10. 2021A Variational Perspective on Diffusion-Based Generative Models and Score Matchingcited by 182
  11. 2016A Kernel Test of Goodness of Fitcited by 143
  12. 2021Enhancing Computational Fluid Dynamics with Machine Learningcited by 137
  13. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 133
  14. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  15. 2019Coarse-scale PDEs from fine-scale observations via machine learningcited by 35
  16. 2019Enforcing Statistical Constraints in Generative Adversarial Networks for Modeling Chaotic Dynamical Systemscited by 33
  17. 2021Discovering Sparse Interpretable Dynamics from Partial Observationscited by 25
  18. 2022Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approachcited by 25
  19. 2020Neural Closure Models for Dynamical Systemscited by 22
  20. 2018Model Reduction with Memory and the Machine Learning of Dynamical Systemscited by 19
  21. 2020Data-driven modelling of nonlinear spatio-temporal fluid flows using a deep convolutional generative adversarial networkcited by 16
  22. 2023Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methodscited by 14
  23. 2023Learning About Structural Errors in Models of Complex Dynamical Systemscited by 13
  24. 2020Numerical methods for nonlocal and fractional modelscited by 12
  25. 2018Enforcing constraints for interpolation and extrapolation in Generative Adversarial Networkscited by 8
  26. 2023Generalized Neural Closure Models with Interpretabilitycited by 8
  27. 2021Time Series Forecasting Using Manifold Learningcited by 7
  28. 2023CEBoosting: Online Sparse Identification of Dynamical Systems with Regime Switching by Causation Entropy Boostingcited by 7
  29. 2023A probabilistic, data-driven closure model for RANS simulations with aleatoric, model uncertaintycited by 7
  30. 2022Comparison of neural closure models for discretised PDEscited by 6
  31. 2024Generative Learning for Forecasting the Dynamics of Complex Systemscited by 6
  32. 2024CGNSDE: Conditional Gaussian Neural Stochastic Differential Equation for Modeling Complex Systems and Data Assimilationcited by 6
  33. 2021ACD-EDMD: Analytical Construction for Dictionaries of Lifting Functions in Koopman Operator-based Nonlinear Robotic Systemscited by 3

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 5; 2026: 5.

12 papers.

  1. 2024Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modelingcited by 36
  2. 2024CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilationcited by 7
  3. 2025Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Networkcited by 6
  4. 2025Active Learning of Model Discrepancy with Bayesian Experimental Designcited by 5
  5. 2025Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximationcited by 5
  6. 2026D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matchingcited by 5
  7. 2025Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Modelscited by 3
  8. 2025Bayesian Experimental Design for Model Discrepancy Calibration: An Auto-Differentiable Ensemble Kalman Inversion Approachcited by 2
  9. 2026Synergizing Transport-Based Generative Models and Latent Geometry for Stochastic Closure Modelingcited by 1
  10. 2026Enforcing Dirichlet Boundary Conditions in Operator Learningcited by 1
  11. 2026Bayesian Experimental Design for Model Discrepancy Calibration: A Rivalry between Kullback--Leibler Divergence and Wasserstein Distancecited by 0
  12. 2026Diffusion-corrected Autoregressive Fourier Neural Operator for Droplet Evolution Predictioncited by 0

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  1. 2025Stochastic generative methods for stable and accurate closure modeling of chaotic dynamical systems
  2. 2026Training-free score-based diffusion for parameter-dependent stochastic dynamical systems
  3. 2025Predicting partially observable dynamical systems via diffusion models with a multiscale inference scheme
  4. 2024Diffusion models as probabilistic neural operators for recovering unobserved states of dynamical systems
  5. 2025Generative AI Models for Learning Flow Maps of Stochastic Dynamical Systems in Bounded Domains
  6. 2024A Training-Free Conditional Diffusion Model for Learning Stochastic Dynamical Systems
  7. 2026Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systems
  8. 2024On conditional diffusion models for PDE simulations