# DeepLag: Discovering Deep Lagrangian Dynamics for Intuitive Fluid Prediction

arXiv: https://arxiv.org/abs/2402.02425 · HTML: https://mlsanity.com/paper/2402.02425/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 57 references that are in the map.
- First posted on arXiv 4 Feb 2024.

## Builds on (most cited first, 29 of 29)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Reformer: The Efficient Transformer](https://mlsanity.com/paper/2001.04451/) (2020)
- [On Layer Normalization in the Transformer Architecture](https://mlsanity.com/paper/2002.04745/) (2020)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Nystr"omformer: A Nystr"om-Based Algorithm for Approximating Self-Attention](https://mlsanity.com/paper/2102.03902/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [SPNets: Differentiable Fluid Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1806.06094/) (2018)
- [Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs](https://mlsanity.com/paper/2109.07143/) (2021)
- [Learning Vortex Dynamics for Fluid Inference and Prediction](https://mlsanity.com/paper/2301.11494/) (2023)

## Built on by (most cited first, 3 of 3)

- [Breaking Scale Anchoring: Frequency Representation Learning for Accurate High-Resolution Inference from Low-Resolution Training](https://mlsanity.com/paper/2512.05132/) (2025)
- [From Fixed Grids to Moving Particles:A Transferable Latent Operator for Fluid Dynamics](https://mlsanity.com/paper/2608.14120/) (2026)
- [Introductory Notes on Learning2](https://mlsanity.com/paper/2609.06546/) (2026)

## Abstract (arXiv)

Accurately predicting the future fluid is vital to extensive areas such as meteorology, oceanology, and aerodynamics. However, since the fluid is usually observed from the Eulerian perspective, its moving and intricate dynamics are seriously obscured and confounded in static grids, bringing thorny challenges to the prediction. This paper introduces a new Lagrangian-Eulerian combined paradigm to tackle the tanglesome fluid dynamics. Instead of solely predicting the future based on Eulerian observations, we propose DeepLag to discover hidden Lagrangian dynamics within the fluid by tracking the movements of adaptively sampled key particles. Further, DeepLag presents a new paradigm for fluid prediction, where the Lagrangian movement of the tracked particles is inferred from Eulerian observations, and their accumulated Lagrangian dynamics information is incorporated into global Eulerian evolving features to guide future prediction respectively. Tracking key particles not only provides a transparent and interpretable clue for fluid dynamics but also makes our model free from modeling complex correlations among massive grids for better efficiency. Experimentally, DeepLag excels in three challenging fluid prediction tasks covering 2D and 3D, simulated and real-world fluids. Code is available at this repository: https://github.com/thuml/DeepLag.