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DeepLag: Discovering Deep Lagrangian Dynamics for Intuitive Fluid Prediction
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Proposes DeepLag, an Eulerian-Lagrangian recurrent network that tracks adaptively sampled key particles alongside a grid-based Eulerian predictor, achieving state-of-the-art fluid prediction on bounded Navier-Stokes, ocean current, and 3D smoke benchmarks.
Problem
Predicting future fluid states from Eulerian grid observations is costly and obscures moving particle dynamics, making long-horizon fluid prediction hard to learn and compute.
Approach
An EuLag Block couples Eulerian grid features with a small set of learnable-sampled Lagrangian particles: particles' dynamics are cross-attended into the Eulerian field, and the evolved Eulerian field is cross-attended back to update particle positions and dynamics, across multiple scales.
Key results
- On Bounded Navier-Stokes, DeepLag reaches relative L2 0.0543 at 10 frames (13.8% better than U-Net's 0.0618) and 0.0993 at 30 frames (2.7% better than LSM).
- On Ocean Current, DeepLag achieves relative L2 0.0168 at 10 days, an 8.3% improvement over the second-best LSM (0.0182).
- On 3D Smoke, DeepLag achieves relative L2 0.0378, a 34.4% improvement over the second-best U-Net (0.0508).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add Lagrangian-guided feature evolving (LagToEu) (ablation) | DeepLag · Bounded Navier-Stokes | Relative L2: 0.0556 -> 0.0543 | Helped |
| Add Eulerian-conditioned particle tracking (EuToLag) (ablation) | DeepLag · Bounded Navier-Stokes | Relative L2: 0.0547 -> 0.0543 | Helped |
| Add learnable particle sampling (ablation) | DeepLag · Bounded Navier-Stokes | Relative L2: 0.0552 -> 0.0543 | Helped |
| Vary particle count to 128, 256, or 768 (instead of 512) (ablation) | DeepLag · Bounded Navier-Stokes | Relative L2: 0.0543; 0.0559; 0.0553; 0.0547 | Mixed |
| Vary spatial scale count to 1, 2, or 5 (instead of 4) (ablation) | DeepLag · Bounded Navier-Stokes | Relative L2: 0.0543; 0.0789; 0.0658; 0.0554 | Mixed |
| Vary latent dimension to 16, 32, or 128 (instead of 64) (ablation) | DeepLag · Bounded Navier-Stokes | Relative L2: 0.0543; 0.0656; 0.0594; 0.0614 | Mixed |
| Swap EuToLag and LagToEu block order (ablation) | DeepLag · Bounded Navier-Stokes (2D) | Relative L2: 0.0543 -> 0.0545 | No clear effect |
| Swap EuToLag and LagToEu block order (ablation) | DeepLag · 3D Smoke (3D) | Relative L2: 0.0378 -> 0.0378 | No clear effect |
| Use DeepLag instead of LSM | Bounded Navier-Stokes, 30 Frames · Bounded Navier-Stokes, 30 Frames | Relative L2: 0.1020 -> 0.0993 | Helped |
| Use DeepLag instead of LSM | Ocean Current, 10 Days · Ocean Current, 10 Days | Relative L2: 0.0182 -> 0.0168 | Helped |
| Use DeepLag instead of LSM | Ocean Current, 30 Days · Ocean Current, 30 Days | Relative L2: 0.0290 -> 0.0257 | Helped |
| Use DeepLag instead of U-Net | 3D Smoke · 3D Smoke | Relative L2: 0.0508 -> 0.0378 | Helped |
Surprising findings
- Swapping the order of the EuToLag and LagToEu attention blocks changes performance only marginally, suggesting the bidirectional information flow is insensitive to order.
- U-Net degenerates severely on 3D Smoke and Transformer-based GNOT and Galerkin Transformer fail there, despite U-Net being competitive on 2D Bounded Navier-Stokes.
Limitations
- Number of tracking particles is a fixed hyperparameter that must be tuned per scenario.
- Lack of Lagrangian supervision in the data means learned Lagrangian dynamics cannot be validated against ground truth.
- Performance gains on the 2D Bounded Navier-Stokes at 30 frames are small (2.7%).
Experimental setup
Datasets: Bounded Navier-Stokes Ocean Current (ECMWF daily sea reanalysis) 3D Smoke
Benchmarks: Bounded Navier-Stokes Ocean Current 3D Smoke
Metrics: relative L2 ACC
Baselines: U-Net FNO Galerkin Transformer Vortex GNOT LSM FactFormer
Builds on: U-Net Galerkin Transformer FactFormer LSM Neural ODE
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