DeepLag: Discovering Deep Lagrangian Dynamics for Intuitive Fluid Prediction

TL;DR

A new Lagrangian-Eulerian combined paradigm to tackle the tanglesome fluid dynamics, 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.

Abstract

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.

Figures (27)

• Figure 1: Comparison between Lagrangian (left) and Eulerian (right) perspectives. The left depicts the learned trajectories of Lagrangian particles overlaid on the mean state, while the right displays the positions of tracked particles in successive Eulerian frames. Fluid motion is more visibly represented through the dynamic Lagrangian view compared to the density variations in static Eulerian grids.
• Figure 2: Three types of neural fluid prediction models (a-c) and overview of DeepLag (d). The EuLag Block accumulates the previous dynamics at each time and scale to guide the Eulerian field update and then evolves the particle movement and dynamics conditioned on the updated field.
• Figure 3: Overview of the EuLag Block, which accumulates previous dynamics information to guide Eulerian evolution for predicting particle movement. Scale index $l$ is omitted for simplicity.
• Figure 4: Showcases (left) and timewise relative L2 (right) on Bounded Navier-Stokes dataset. Both predictions (upper row) and absolute error maps (lower row) are plotted for intuitive comparison.
• Figure 5: Showcase comparison and visualization of Lagrangian trajectories learned by DeepLag on Ocean Current. Notably, potential temperatures predicted by different models are plotted. Error maps of predictions are normalized to $(-4,4)$ for a better view.