From Fixed Grids to Moving Particles:A Transferable Latent Operator for Fluid Dynamics
Authors: Meng Li, Chuqi Chen, Zhengqing Gao, Xi Zhou, Xiao Sun, Yang Xiang, Huaxi Huang
Organizations: Shanghai Artificial Intelligence Laboratory · The Hong Kong University of Science and Technology · University of Michigan · Mohamed bin Zayed University of Artificial Intelligence · HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute
Lagrangian modeling is vital to fluid dynamics, as it characterizes particle transport and complements the Eulerian representation. However, Lagrangian trajectories are less commonly available than Eulerian fields, while most neural operators are trained and evaluated primarily in the Eulerian representation. This mismatch motivates a new learning problem: can a model trained solely on Eulerian observations generalize zero-shot from Eulerian field prediction to Lagrangian particle rollout, without Lagrangian supervision or task-specific adaptation? To address this problem, we propose the Transferable Latent Operator (TLO), which learns a unified flow representation shared by Eulerian field prediction and Lagrangian particle rollout. TLO decouples latent flow evolution from coordinate-dependent decoding: querying the evolving latent representation at fixed spatial coordinates yields Eulerian fields, whereas querying velocities at particle positions and recursively updating these positions enables Lagrangian rollout. Across five fluid-dynamics benchmarks, TLO consistently outperforms existing neural operators in both Eulerian field prediction and zero-shot Lagrangian rollout, with further gains from limited Lagrangian fine-tuning.
Figures & tables
Figure 1: Workflow of TLO for unified Eulerian field forecasting and Lagrangian particle rollout from a shared latent flow representation. Notations XE , U^t , and y^t follow Section Problem Setting.
Figure 2: Overview of the Transferable Latent Operator ( TLO ). TLO learns coordinate-independent latent dynamics from Eulerian observations. The hierarchical encoder extracts local and global flow representations, while the latent processor evolves the fused tokens without spatial coordinates. A coordinate-conditioned decoder then queries the latent state at arbitrary locations, enabling both fixed-grid Eulerian prediction and Lagrangian rollout. Particle positions are used only as inference-time queries and are not required for Eulerian training.
Dataset
Native data
Support
Evaluation
Horizon
DAM2D
SPH
32×64
Eul/Ref/Path
10
NS2D
Grid
642
Eul
10
TGV3D
SPH
203
Eul
5
Burgers3D
Grid
323
Eul/Ref/Path
5
SEA
Grid
180×300
Eul/Ref/Path
10
Table 1: Benchmark summary. Eul, Ref, and Path denote fixed-grid field, reference-query velocity, and closed-loop trajectory evaluation.
Method
DAM2D
NS2D
TGV3D
Burgers3D
SEA
LSM
2.97E-02
3.46E-03
9.57E-04
2.17E-03
5.72E-01
GINO
6.04E-03
3.68E-05
2.47E-04
5.35E-03
5.90E-01
GNOT
2.24E-02
1.72E-03
9.37E-04
4.21E-04
4.19E-01
Transolver
6.13E-03
1.40E-04
7.67E-04
4.22E-04
5.22E-01
LNO
3.75E-03
8.73E-05
7.54E-03
4.42E-03
5.21E-01
UPT
2.38E-02
1.31E-03
7.54E-03
5.59E-03
8.97E-01
Table 2: Mean squared error (MSE) of autoregressive Eulerian rollout. Lower is better. The best result in each column is highlighted in bold.
Figure 3: Patch and token ablations on NS2D. Left: Eul MSE versus patch lattice size. Right: two models use matched total tokens, with a fixed 8×8 patch and global token equals 8 for Local+Global. Lower is better.
Design
Eul ↓
TLO (Local 56 + Global 8)
1.98E-05
Local-only (64)
2.22E-05
Global-only (64)
4.21E-05
w/o FFN
4.51E-05
Table 3: Ablation of TLO components on NS2D under comparable model sizes. Token-allocation variants share the same 64-token budget. Lower L2 error is better.
Dataset
Setting
Method
Eul ↓
Ref ↓
Path ↓
DAM2D
EO
UPT
2.38E-02
7.62E-02
1.44E-01
LNO
3.75E-03
4.35E-03
5.30E-03
TLO (Ours)
3.12E-03
2.41E-03
3.72E-03
DA
TLO (Ours)
3.08E-03
2.39E-03
3.72E-03
Burgers3D
EO
UPT
5.59E-03
2.91E-03
7.73E-03
LNO
4.42E-03
3.36E-03
9.42E-03
Table 4: Transfer from Eulerian training to Lagrangian rollout. EO directly applies the Eulerian-trained checkpoint without particle-based training supervision. For TLO , DA fine-tunes only the decoder using sparse velocity samples at particle coordinates. Eul, Ref, and Path denote fixed-grid field, reference-path velocity, and closed-loop particle-position MSE, respectively. Lower is better.
Figure 4: Qualitative closed-loop Lagrangian rollout on SEA. Each column shows the Eulerian velocity field and the corresponding particle trajectories at different rollout steps ( T=2,6,10 ).
Method
Ref ↓
Path ↓
Mem. ↓
Time ↓
TLO-DA
2.39E-03
3.72E-03
35.94
7.26
GNS
4.19E-02
2.50E-02
47.99
8.68
Table 5: Comparison with a native Lagrangian model on DAM2D particle rollout. TLO and GNS contain 144K and 146K parameters, respectively. Memory and runtime are reported in MiB and ms/step, respectively.
In-grid 10 steps
Outside 20 steps
Full 20 steps
Method
Ref
Path
Ref
Path
Path
TLO -Interp
2.89E-04
1.63E-04
–
–
–
TLO -Direct
3.22E-04
1.89E-04
1.35E-01
6.05E-02
4.48E-03
TLO -DA
3.22E-04
1.89E-04
3.82E-02
3.51E-02
2.51E-03
Table 6: Direct decoding and grid interpolation on DAM2D. A dash indicates that interpolation cannot continue after grid exit.
Figure 5: Closed-loop rollout beyond the fixed Eulerian grid support on DAM2D. Light- and dark-blue points show reference particles at Steps 15 and 20, respectively. The inset follows a particle crossing Ωgrid : grid interpolation becomes undefined after the exit, whereas direct decoding continues the rollout.