# Bayesian Conditional Diffusion Models for Versatile Spatiotemporal Turbulence Generation

arXiv: https://arxiv.org/abs/2311.07896 · HTML: https://mlsanity.com/paper/2311.07896/

## Key facts

- Cited by 29 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 26 of its 59 references that are in the map.
- First posted on arXiv 14 Nov 2023.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 26 of 26)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022)
- [A note on the evaluation of generative models](https://mlsanity.com/paper/1511.01844/) (2015)
- [A Review on Generative Adversarial Networks: Algorithms, Theory, and Applications](https://mlsanity.com/paper/2001.06937/) (2020)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Seeing What a GAN Cannot Generate](https://mlsanity.com/paper/1910.11626/) (2019)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics](https://mlsanity.com/paper/2205.03990/) (2022)
- [Debias Coarsely, Sample Conditionally: Statistical Downscaling through Optimal Transport and Probabilistic Diffusion Models](https://mlsanity.com/paper/2305.15618/) (2023)
- [Generative Modeling of Turbulence](https://mlsanity.com/paper/2112.02548/) (2021)
- [Multi-fidelity Generative Deep Learning Turbulent Flows](https://mlsanity.com/paper/2006.04731/) (2020)
- [Differentiable hybrid neural modeling for fluid-structure interaction](https://mlsanity.com/paper/2303.12971/) (2023)
- [Deep Learning of Turbulent Scalar Mixing](https://mlsanity.com/paper/1811.07095/) (2018)
- [Reconstruction of turbulent data with deep generative models for semantic inpainting from TURB-Rot database](https://mlsanity.com/paper/2006.09179/) (2020)
- [Assessments of epistemic uncertainty using Gaussian stochastic weight averaging for fluid-flow regression](https://mlsanity.com/paper/2109.08248/) (2021)
- [Diffusion model based data generation for partial differential equations](https://mlsanity.com/paper/2306.11075/) (2023)

## Built on by (most cited first, 29 of 29)

- [Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networks](https://mlsanity.com/paper/2504.02843/) (2025)
- [FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modeling](https://mlsanity.com/paper/2506.07902/) (2025)
- [Generative Learning of the Solution of Parametric Partial Differential Equations Using Guided Diffusion Models and Virtual Observations](https://mlsanity.com/paper/2408.00157/) (2024)
- [Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator](https://mlsanity.com/paper/2408.02965/) (2024)
- [Diffusion-Based Hierarchical Graph Neural Networks for Simulating Nonlinear Solid Mechanics](https://mlsanity.com/paper/2506.06045/) (2025)
- [Diff-SPORT: Diffusion-based Sensor Placement Optimization and Reconstruction of Turbulent flows in urban environments](https://mlsanity.com/paper/2506.00214/) (2025)
- [Learning Physics for Unveiling Hidden Earthquake Ground Motions via Conditional Generative Modeling](https://mlsanity.com/paper/2407.15089/) (2024)
- [Generative Learning for Forecasting the Dynamics of Complex Systems](https://mlsanity.com/paper/2402.17157/) (2024)
- [Generative prediction of flow fields around an obstacle using the diffusion model](https://mlsanity.com/paper/2407.00735/) (2024)
- [D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matching](https://mlsanity.com/paper/2602.21469/) (2026)
- [Rectified Flows for Fast Multiscale Fluid Flow Modeling](https://mlsanity.com/paper/2506.03111/) (2025)
- [Model-Agnostic AI Framework with Explicit Time Integration for Long-Term Fluid Dynamics Prediction](https://mlsanity.com/paper/2412.05657/) (2024)
- [CD-TVD: Contrastive Diffusion for 3D Super-Resolution with Scarce High-Resolution Time-Varying Data](https://mlsanity.com/paper/2508.08173/) (2025)
- [Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systems](https://mlsanity.com/paper/2603.12635/) (2026)
- [Multi-fidelity Reinforcement Learning Control for Complex Dynamical Systems](https://mlsanity.com/paper/2504.05588/) (2025)
- [Generative forecasting with joint probability models](https://mlsanity.com/paper/2512.24446/) (2025)
- [Comparison of Generative Learning Methods for Turbulence Surrogates](https://mlsanity.com/paper/2411.16417/) (2024)
- [Control-Augmented Autoregressive Diffusion for Data Assimilation](https://mlsanity.com/paper/2510.06637/) (2025)
- [Synergizing Transport-Based Generative Models and Latent Geometry for Stochastic Closure Modeling](https://mlsanity.com/paper/2602.17089/) (2026)
- [FoilDiff: A Hybrid Transformer Backbone for Diffusion-based Modelling of 2D Airfoil Flow Fields](https://mlsanity.com/paper/2510.04325/) (2025)
- [Learning Temporally Consistent Turbulence Between Sparse Snapshots via Diffusion Models](https://mlsanity.com/paper/2512.24813/) (2025)
- [Coupled Diffusion-Encoder Models for Reconstruction of Flow Fields](https://mlsanity.com/paper/2601.07946/) (2026)
- [Learning noisy phase transition dynamics from stochastic partial differential equations](https://mlsanity.com/paper/2604.09664/) (2026)
- [Data Enrichment for Symbolic Regression Using Diffusion Models](https://mlsanity.com/paper/2606.00988/) (2026)
- [Multiscale Hypersonic Boundary Layer Reconstruction via Spectral Binning and Subdomain-wise Conditional Diffusion](https://mlsanity.com/paper/2606.15023/) (2026)
- [Patched Flow Matching: Generative Wall-Pressure Reconstruction Beyond Training-Domain Scales from Sparse Sensors](https://mlsanity.com/paper/2606.22084/) (2026)
- [A Shortcut to Statistically Steady-State Turbulence with Flow Matching](https://mlsanity.com/paper/2607.13022/) (2026)
- [MoRF-AST: Calibrated Probabilistic Virtual Sensing for Structural Monitoring under Changing Operating Conditions](https://mlsanity.com/paper/2608.24531/) (2026)
- [Conditional Flow Matching for Generation of 3D Multi-variable Instantaneous Urban Microclimate Fields](https://mlsanity.com/paper/2610.10430/) (2026)

## Abstract (arXiv)

Turbulent flows have historically presented formidable challenges to predictive computational modeling. Traditional numerical simulations often require vast computational resources, making them infeasible for numerous engineering applications. As an alternative, deep learning-based surrogate models have emerged, offering data-drive solutions. However, these are typically constructed within deterministic settings, leading to shortfall in capturing the innate chaotic and stochastic behaviors of turbulent dynamics. We introduce a novel generative framework grounded in probabilistic diffusion models for versatile generation of spatiotemporal turbulence. Our method unifies both unconditional and conditional sampling strategies within a Bayesian framework, which can accommodate diverse conditioning scenarios, including those with a direct differentiable link between specified conditions and generated unsteady flow outcomes, and scenarios lacking such explicit correlations. A notable feature of our approach is the method proposed for long-span flow sequence generation, which is based on autoregressive gradient-based conditional sampling, eliminating the need for cumbersome retraining processes. We showcase the versatile turbulence generation capability of our framework through a suite of numerical experiments, including: 1) the synthesis of LES simulated instantaneous flow sequences from URANS inputs; 2) holistic generation of inhomogeneous, anisotropic wall-bounded turbulence, whether from given initial conditions, prescribed turbulence statistics, or entirely from scratch; 3) super-resolved generation of high-speed turbulent boundary layer flows from low-resolution data across a range of input resolutions. Collectively, our numerical experiments highlight the merit and transformative potential of the proposed methods, making a significant advance in the field of turbulence generation.