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Learning Reduced Fluid Dynamics
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Learns a low-dimensional, exactly time-reversible reduced model of incompressible inviscid Eulerian fluids by optimizing divergence-free POD-initialized bases on the Grassmannian/Stiefel manifold with stochastic Riemannian optimization against a trajectory-wise discrepancy loss.
Problem
Predicting long-horizon state evolution of ultra high-dimensional, time-reversible fluid dynamic systems is computationally expensive, and existing non-intrusive or PINN-based learned models either are costly or fail to preserve time reversibility and energy.
Approach
Treats the reduced velocity subspace basis as the decision variable, lifts reduced trajectories to the ambient space so they are comparable across bases, proves the trajectory map is differentiable over the Grassmannian under small timesteps, and uses tensor precomputation so the trajectory-wise discrepancy gradient is computed without full-order time integration.
Key results
- Reduces trajectory-wise discrepancy loss over POD by 87.93%-91.47% on Taylor vortices across four bases sizes.
- On the rising smoke plume benchmark, reduces discrepancy by 80.94%-88.82% across four settings relative to POD.
- On held-out spherical plume test angles, reduces discrepancy by 54.65% on average; on two-plume test set by 59.28% on average.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Fine-tune POD bases with the proposed trajectory-wise optimization | Reduced fluid model · Taylor Vortices | discrepancy loss: 87.93%, 90.12%, 91.47%, and 90.16% | Helped |
| Fine-tune POD bases with the proposed trajectory-wise optimization | Reduced fluid model · Plume Rise | discrepancy loss: 88.82%, 80.94%, 87.60%, and 75.79% | Helped |
| Fine-tune POD bases with the proposed trajectory-wise optimization | Reduced fluid model · Plume Rise+Obstacle | discrepancy loss: 92.13%, 81.49%, 85.38%, and 81.70% | Helped |
| Fine-tune POD bases with the proposed trajectory-wise optimization | Reduced fluid model · Spherical Plume | discrepancy: 54.65% | Helped |
| Fine-tune POD bases with the proposed trajectory-wise optimization | Reduced fluid model · Two Plume | discrepancy: 59.28% | Helped |
| Use full-unrolling rather than one-step training (ablation) | Reduced fluid model · Plume Rise, ϵ = 0.05, p = 6 | discrepancy loss after 3k iterations: 57.39 -> 6.37; 24.12 | Helped |
| Use full-unrolling rather than one-step training (ablation) | Reduced fluid model · Plume Rise, ϵ = 0.01, p = 9 | discrepancy loss after 3k iterations: 28.23 -> 5.38; 15.71 | Helped |
| Increase the initialization noise level from ˜ϵ = 0.01 to ˜ϵ = 0.5 (ablation) | Reduced fluid model on Taylor Vortices · Taylor Vortices, ϵ = 0.05, p = 8 | initial discrepancy loss: 4.84 -> 22.77 | Hurt |
| Increase the initialization noise level from ˜ϵ = 0.01 to ˜ϵ = 0.5 (ablation) | Reduced fluid model on Taylor Vortices · Taylor Vortices, ϵ = 0.05, p = 8 | discrepancy loss after 8k iterations: 0.79 -> 1.51 | Hurt |
Surprising findings
- A fully noisy random initialization of the basis leads to a much worse final result than POD initialization, so the POD baseline is a useful starting point.
Limitations
- Training takes hours to days and trajectory prediction is sequential, so it cannot exploit GPU parallelism.
- Linear subspace has limited expressivity compared with neural network reduced models.
- Cannot model free-surface flows or non-reversible (dissipative) flows without additional terms.
Experimental setup
Datasets: simulated trajectories from full-order Eulerian fluid model (Taylor vortices, smoke plume benchmarks)
Benchmarks: Taylor vortices rising smoke plume plume rise with obstacle spherical plume two plumes
Metrics: trajectory-wise discrepancy loss energy conservation runtime
Baselines: POD Dynamic Mode Decomposition PINN-style physics loss one-step training (Brandstetter et al. 2022)
Builds on: Proper Orthogonal Decomposition Galerkin projection reduced-order model (Liu et al. 2015) Pavlov et al. 2011 time-reversible discretization RAMSGrad Physics-informed loss (Sirignano and Spiliopoulos 2018)
Closest papers by meaning
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