# Synthetic Lagrangian Turbulence by Generative Diffusion Models

arXiv: https://arxiv.org/abs/2307.08529 · HTML: https://mlsanity.com/paper/2307.08529/

## Key facts

- Cited by 32 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 19 of its 94 references that are in the map.
- First posted on arXiv 17 Jul 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, 19 of 19)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Wavelet Score-Based Generative Modeling](https://mlsanity.com/paper/2208.05003/) (2022)
- [Convolutional-network models to predict wall-bounded turbulence from wall quantities](https://mlsanity.com/paper/2006.12483/) (2020)
- [Accurate and Conservative Estimates of MRF Log-likelihood using Reverse Annealing](https://mlsanity.com/paper/1412.8566/) (2014)
- [Reconstruction of turbulent data with deep generative models for semantic inpainting from TURB-Rot database](https://mlsanity.com/paper/2006.09179/) (2020)
- [Optimal tracking strategies in a turbulent flow](https://mlsanity.com/paper/2305.04677/) (2023)
- [Neural network based generation of a 1-dimensional stochastic field with turbulent velocity statistics](https://mlsanity.com/paper/2211.11580/) (2022)

## Built on by (most cited first, 30 of 32)

- [Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training](https://mlsanity.com/paper/2505.17638/) (2025)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [Generative Latent Neural PDE Solver using Flow Matching](https://mlsanity.com/paper/2503.22600/) (2025)
- [Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints](https://mlsanity.com/paper/2506.08604/) (2025)
- [Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator](https://mlsanity.com/paper/2408.02965/) (2024)
- [PiRD: Physics-informed Residual Diffusion for Flow Field Reconstruction](https://mlsanity.com/paper/2404.08412/) (2024)
- [Turbulence Scaling from Deep Learning Diffusion Generative Models](https://mlsanity.com/paper/2311.06112/) (2023)
- [Diff-SPORT: Diffusion-based Sensor Placement Optimization and Reconstruction of Turbulent flows in urban environments](https://mlsanity.com/paper/2506.00214/) (2025)
- [The wall confronting large language models](https://mlsanity.com/paper/2507.19703/) (2025)
- [Generative prediction of flow fields around an obstacle using the diffusion model](https://mlsanity.com/paper/2407.00735/) (2024)
- [DiffBatt: A Diffusion Model for Battery Degradation Prediction and Synthesis](https://mlsanity.com/paper/2410.23893/) (2024)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Accurate and scalable deep Maxwell solvers using multilevel iterative methods](https://mlsanity.com/paper/2509.03622/) (2025)
- [Understanding complex crowd dynamics with generative neural simulators](https://mlsanity.com/paper/2412.01491/) (2024)
- [From Uncertain to Safe: Conformal Adaptation of Diffusion Models for Safe PDE Control](https://mlsanity.com/paper/2502.02205/) (2025)
- [Data-driven particle dynamics: Structure-preserving coarse-graining for emergent behavior in non-equilibrium systems](https://mlsanity.com/paper/2508.12569/) (2025)
- [A multiscale and multicriteria Generative Adversarial Network to synthesize 1-dimensional turbulent fields](https://mlsanity.com/paper/2307.16580/) (2023)
- [The Universal Statistical Structure and Scaling Laws of Chaos and Turbulence](https://mlsanity.com/paper/2311.01358/) (2023)
- [PODiff: Latent Diffusion in Proper Orthogonal Decomposition Space for Scientific Super-Resolution](https://mlsanity.com/paper/2605.03399/) (2026)
- [Least-Action-Guided Diffusion for Physical Extrapolation](https://mlsanity.com/paper/2606.11277/) (2026)
- [Stochastic Reconstruction of Gappy Lagrangian Turbulent Signals by Conditional Diffusion Models](https://mlsanity.com/paper/2410.23971/) (2024)
- [Capturing non-Markovian dynamics in non-equilibrium stochastic systems using flow matching](https://mlsanity.com/paper/2606.06658/) (2026)
- [Mixing Artificial and Natural Intelligence: From Statistical Mechanics to AI and Back to Turbulence](https://mlsanity.com/paper/2403.17993/) (2024)
- [Slim multi-scale convolutional autoencoder-based reduced-order models for interpretable features of a complex dynamical system](https://mlsanity.com/paper/2501.03070/) (2025)
- [Data-driven Mori-Zwanzig modeling of Lagrangian particle dynamics in turbulent flows](https://mlsanity.com/paper/2507.16058/) (2025)
- [Deep Generative Model for Human Mobility Behavior](https://mlsanity.com/paper/2510.06473/) (2025)
- [Reconstructing Multi-Scale Physical Fields from Extremely Sparse Measurements with an Autoencoder-Diffusion Cascade](https://mlsanity.com/paper/2512.01572/) (2025)
- [Learning Temporally Consistent Turbulence Between Sparse Snapshots via Diffusion Models](https://mlsanity.com/paper/2512.24813/) (2025)
- [Explainable deep learning reveals the physical mechanisms behind the turbulent kinetic energy equation](https://mlsanity.com/paper/2601.20052/) (2026)
- [Annealed Co-Generation: Disentangling Variables via Progressive Pairwise Modeling](https://mlsanity.com/paper/2603.06615/) (2026)

## Abstract (arXiv)

Lagrangian turbulence lies at the core of numerous applied and fundamental problems related to the physics of dispersion and mixing in engineering, bio-fluids, atmosphere, oceans, and astrophysics. Despite exceptional theoretical, numerical, and experimental efforts conducted over the past thirty years, no existing models are capable of faithfully reproducing statistical and topological properties exhibited by particle trajectories in turbulence. We propose a machine learning approach, based on a state-of-the-art diffusion model, to generate single-particle trajectories in three-dimensional turbulence at high Reynolds numbers, thereby bypassing the need for direct numerical simulations or experiments to obtain reliable Lagrangian data. Our model demonstrates the ability to reproduce most statistical benchmarks across time scales, including the fat-tail distribution for velocity increments, the anomalous power law, and the increased intermittency around the dissipative scale. Slight deviations are observed below the dissipative scale, particularly in the acceleration and flatness statistics. Surprisingly, the model exhibits strong generalizability for extreme events, producing events of higher intensity and rarity that still match the realistic statistics. This paves the way for producing synthetic high-quality datasets for pre-training various downstream applications of Lagrangian turbulence.