mlsanityData to 7 Oct 2026

Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modeling

arXiv 2409.08477 ↗First posted 13 Sep 2024cs.LG cs.AI physics.flu-dyncited by 36

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

Abstract (arXiv)

We integrate neural operators with diffusion models to address the spectral limitations of neural operators in surrogate modeling of turbulent flows. While neural operators offer computational efficiency, they exhibit deficiencies in capturing high-frequency flow dynamics, resulting in overly smooth approximations. To overcome this, we condition diffusion models on neural operators to enhance the resolution of turbulent structures. Our approach is validated for different neural operators on diverse datasets, including a high Reynolds number jet flow simulation and experimental Schlieren velocimetry. The proposed method significantly improves the alignment of predicted energy spectra with true distributions compared to neural operators alone. This enables the diffusion models to stabilize longer forecasts through diffusion-corrected autoregressive rollouts, as we demonstrate in this work. Additionally, proper orthogonal decomposition analysis demonstrates enhanced spectral fidelity in space-time. This work establishes a new paradigm for combining generative models with neural operators to advance surrogate modeling of turbulent systems, and it can be used in other scientific applications that involve microstructure and high-frequency content. See our project page: vivekoommen.github.io/NO_DM

Key facts. Cited by 36 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 31 of its 55 references that are in the map. First posted on arXiv 13 Sep 2024.

Builds on

31 of 55 references.

Closest ancestors · 4

Same direction, newest first

  1. 2024Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator6 Aug
  2. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observation25 Jun
  3. 2023Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulation4 Sep
  4. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers10 Aug

Other references · 27

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  3. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  5. 2022Elucidating the Design Space of Diffusion-Based Generative Modelscited by 2,994
  6. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  7. 2015Variational Inference with Normalizing Flowscited by 2,161
  8. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  9. 2021Palette: Image-to-Image Diffusion Modelscited by 1,296
  10. 2022Diffusion Models: A Comprehensive Survey of Methods and Applicationscited by 1,107
  11. 2022Diffusion Models in Vision: A Surveycited by 1,008
  12. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 321
15 more
  1. 2022Transformer for Partial Differential Equations' Operator Learningcited by 270
  2. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
  3. 2020Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domainscited by 78
  4. 2022Learning two-phase microstructure evolution using neural operators and autoencoder architecturescited by 72
  5. 2023Residual-based attention in physics-informed neural networkscited by 72
  6. 2021Physics-based Deep Learningcited by 67
  7. 2023AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identificationcited by 29
  8. 2024RiemannONets: Interpretable Neural Operators for Riemann Problemscited by 28
  9. 2023Discovering a reaction-diffusion model for Alzheimer's disease by combining PINNs with symbolic regressioncited by 25
  10. 2023Rethinking materials simulations: Blending direct numerical simulations with neural operatorscited by 25
  11. 2022Neural Integral Equationscited by 23
  12. 2023A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Datacited by 18
  13. 2023Learning bias corrections for climate models using deep neural operatorscited by 9
  14. 2023AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regressioncited by 9
  15. 2023On the locality of local neural operator in learning fluid dynamicscited by 6

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 5; 2025: 17; 2026: 14.

36 papers.

Sort:

By area:

  1. 2025Mitigating Spectral Bias in Neural Operators via High-Frequency Scaling for Physical Systemscited by 29
  2. 2025FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modelingcited by 19
  3. 2024Virtual Sensing-Enabled Digital Twin Framework for Real-Time Monitoring of Nuclear Systems Leveraging Deep Neural Operatorscited by 12
  4. 2024Predicting Crack Nucleation and Propagation in Brittle Materials Using Deep Operator Networks with Diverse Trunk Architecturescited by 12
  5. 2025Binned Spectral Power Loss for Improved Prediction of Chaotic Systemscited by 10
  6. 2024Predicting Change, Not States: An Alternate Framework for Neural PDE Surrogatescited by 8
  7. 2025FLEX: A Backbone for Diffusion-Based Modeling of Spatio-temporal Physical Systemscited by 8
  8. 2025FourierFlow: Frequency-aware Flow Matching for Generative Turbulence Modelingcited by 8
  9. 2025Generative emulation of chaotic dynamics with coherent priorcited by 7
  10. 2025A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learningcited by 6
  11. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  12. 2025Rectified Flows for Fast Multiscale Fluid Flow Modelingcited by 4
24 more
  1. 2026(Sparse) Attention to the Details: Preserving Spectral Fidelity in ML-based Weather Forecasting Modelscited by 4
  2. 2026FlowRefiner: Flow Matching-Based Iterative Refinement for 3D Turbulent Flow Simulationcited by 4
  3. 2025Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutionscited by 3
  4. 2025Flow marching for a generative PDE foundation modelcited by 3
  5. 2026Physics-informed diffusion models in spectral spacecited by 3
  6. 2026Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systemscited by 3
  7. 2026MENO: MeanFlow-Enhanced Neural Operators for Dynamical Systemscited by 3
  8. 2025Equilibrium Conserving Neural Operators for Super-Resolution Learningcited by 2
  9. 2025Reframing Generative Models for Physical Systems using Stochastic Interpolantscited by 2
  10. 2026Divergence-Free Diffusion Models for Incompressible Fluid Flowscited by 2
  11. 2026Latent Generative Solvers for Generalizable Long-Term Physics Simulationcited by 2
  12. 2024Implicit factorized transformer approach to fast prediction of turbulent channel flowscited by 1
  13. 2025PolyMicros: Bootstrapping a Foundation Model for Polycrystalline Material Structurecited by 1
  14. 2025Importance of localized dilatation and distensibility in identifying determinants of thoracic aortic aneurysm with neural operatorscited by 1
  15. 2024Variational autoencoders with latent high-dimensional steady geometric flows for dynamicscited by 0
  16. 2025DETNO: A Diffusion-Enhanced Transformer Neural Operator for Long-Term Traffic Forecastingcited by 0
  17. 2025Reconstructing Multi-Scale Physical Fields from Extremely Sparse Measurements with an Autoencoder-Diffusion Cascadecited by 0
  18. 2025Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equationscited by 0
  19. 2026Autoregressive One-Step Generative Modeling for Dynamical System Forecastingcited by 0
  20. 2026NOFE - Neural Operator Function Embeddingcited by 0
  21. 2026Wavelet Flow Matching for Multi-Scale Physics Emulationcited by 0
  22. 2026Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observationscited by 0
  23. 2026Spectrally Regularized Latent Flow Matching for Turbulence Generationcited by 0
  24. 2026The Singularity Space: A Generative Diffusion Framework for Signal Representationcited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2025Resolving Turbulent Magnetohydrodynamics: A Hybrid Operator-Diffusion Framework
  2. 2025CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operators
  3. 2026Learning to Discretize: Diffusion-Based Adaptive Mesh with Spectral Guidance
  4. 2025Guiding diffusion models to reconstruct flow fields from sparse data
  5. 2025Lazy Diffusion: Mitigating spectral collapse in generative diffusion-based stable autoregressive emulation of turbulent flows
  6. 2023Turbulence Scaling from Deep Learning Diffusion Generative Models
  7. 2024Unfolding Time: Generative Modeling for Turbulent Flows in 4D
  8. 2024CoNFiLD-inlet: Synthetic Turbulence Inflow Using Generative Latent Diffusion Models with Neural Fields