# Neural Ideal Large Eddy Simulation: Modeling Turbulence with Neural Stochastic Differential Equations

arXiv: https://arxiv.org/abs/2306.01174 · HTML: https://mlsanity.com/paper/2306.01174/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 82 references that are in the map.
- First posted on arXiv 1 Jun 2023.

## Builds on (most cited first, 27 of 27)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Multiscale Vision Transformers](https://mlsanity.com/paper/2104.11227/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [MViTv2: Improved Multiscale Vision Transformers for Classification and Detection](https://mlsanity.com/paper/2112.01526/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [GraphCast: Learning skillful medium-range global weather forecasting](https://mlsanity.com/paper/2212.12794/) (2022)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Pangu-Weather: A 3D High-Resolution Model for Fast and Accurate Global Weather Forecast](https://mlsanity.com/paper/2211.02556/) (2022)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [Learning to correct spectral methods for simulating turbulent flows](https://mlsanity.com/paper/2207.00556/) (2022)
- [Learned Turbulence Modelling with Differentiable Fluid Solvers: Physics-based Loss-functions and Optimisation Horizons](https://mlsanity.com/paper/2202.06988/) (2022)
- [Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equations](https://mlsanity.com/paper/2102.06559/) (2021)
- [A Perspective on Machine Learning Methods in Turbulence Modelling](https://mlsanity.com/paper/2010.12226/) (2020)
- [Data-driven Discovery of Closure Models](https://mlsanity.com/paper/1803.09318/) (2018)
- [A machine learning framework for data driven acceleration of computations of differential equations](https://mlsanity.com/paper/1807.09519/) (2018)
- [Embedded training of neural-network sub-grid-scale turbulence models](https://mlsanity.com/paper/2105.01030/) (2021)
- [Long-term stability and generalization of observationally-constrained stochastic data-driven models for geophysical turbulence](https://mlsanity.com/paper/2205.04601/) (2022)
- [Computing foaming flows across scales: from breaking waves to microfluidics](https://mlsanity.com/paper/2103.01513/) (2021)

## Built on by (most cited first, 13 of 13)

- [DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems](https://mlsanity.com/paper/2402.04467/) (2024)
- [Uncertainty-aware Surrogate Models for Airfoil Flow Simulations with Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2312.05320/) (2023)
- [Efficient, Accurate and Stable Gradients for Neural ODEs](https://mlsanity.com/paper/2410.11648/) (2024)
- [EddyFormer: Accelerated Neural Simulations of Three-Dimensional Turbulence at Scale](https://mlsanity.com/paper/2510.24173/) (2025)
- [A probabilistic framework for learning non-intrusive corrections to long-time climate simulations from short-time training data](https://mlsanity.com/paper/2408.02688/) (2024)
- [Learning Stochastic Multiscale Models](https://mlsanity.com/paper/2506.22655/) (2025)
- [AutoHood3D: A Multi-Modal Benchmark for Automotive Hood Design and Fluid-Structure Interaction](https://mlsanity.com/paper/2511.05596/) (2025)
- [Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling](https://mlsanity.com/paper/2510.00233/) (2025)
- [Learning Permutation-invariant Macroscopic Dynamics](https://mlsanity.com/paper/2605.30812/) (2026)
- [Attention-enhanced neural differential equations for physics-informed deep learning of ion transport](https://mlsanity.com/paper/2312.02871/) (2023)
- [Scalable learning of macroscopic stochastic dynamics](https://mlsanity.com/paper/2511.12842/) (2025)
- [Improving Infinitely Deep Bayesian Neural Networks with Nesterov's Accelerated Gradient Method](https://mlsanity.com/paper/2603.25024/) (2026)
- [Flux-form spatiotemporal neural operators for coarse-grained dynamics of multiscale PDEs](https://mlsanity.com/paper/2608.18148/) (2026)

## Abstract (arXiv)

We introduce a data-driven learning framework that assimilates two powerful ideas: ideal large eddy simulation (LES) from turbulence closure modeling and neural stochastic differential equations (SDE) for stochastic modeling. The ideal LES models the LES flow by treating each full-order trajectory as a random realization of the underlying dynamics, as such, the effect of small-scales is marginalized to obtain the deterministic evolution of the LES state. However, ideal LES is analytically intractable. In our work, we use a latent neural SDE to model the evolution of the stochastic process and an encoder-decoder pair for transforming between the latent space and the desired ideal flow field. This stands in sharp contrast to other types of neural parameterization of closure models where each trajectory is treated as a deterministic realization of the dynamics. We show the effectiveness of our approach (niLES - neural ideal LES) on a challenging chaotic dynamical system: Kolmogorov flow at a Reynolds number of 20,000. Compared to competing methods, our method can handle non-uniform geometries using unstructured meshes seamlessly. In particular, niLES leads to trajectories with more accurate statistics and enhances stability, particularly for long-horizon rollouts.