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Lagrangian Flow Networks for Conservation Laws
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Proposes Lagrangian Flow Networks (LFlows), conditional normalizing flows whose time-conditioned bijections yield densities and velocities that satisfy the continuity equation by construction, evaluated on simulated fluid flow, dynamical optimal transport, and bird migration from radar data.
Problem
Modeling spatiotemporal fluid densities and velocities from sparse, noisy data (with unknown initial/boundary conditions) while exactly satisfying mass conservation (the continuity equation).
Approach
Model the density at every time as a time-conditioned normalizing flow transformation of a fixed base density, so the continuity equation holds by construction; the velocity is obtained analytically from the forward map via the Jacobian, avoiding inversion of the bijections and numerical ODE solvers.
Key results
- LFlows show the highest average R^2 for density in both 2D and 3D synthetic fluid flow, while PINNs degrade severely in 3D.
- All methods enforcing the continuity equation (LFlows, SLDA, DFNNs) achieve consistency sMAPE on the order of 1e-4 or lower, versus the PINN's inconsistency.
- In dynamical OT, LFlow W2^2 estimates are closest to the discrete OT reference range, while the minimax ICNN underestimates and DFNNs overestimate.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use LFlows instead of competing methods | density prediction on simulated fluid flow · Simulated fluid flow (2D and 3D) | R² | Helped |
| use PINNs instead of LFlows | density prediction on simulated fluid flow · Simulated fluid flow (3D) | R² | Hurt |
| use LFlows instead of the baseline MLP | bird migration density prediction · Bird migration | test density error | Helped |
| use LFlows instead of DFNNs or the minimax model | optimal transport estimation · Dynamical optimal transport: Circles ↔ Pinwheel, Pinwheel ↔ 8Gaussians, 8Gaussians ↔ Circles | W₂² estimate closeness to discrete estimates | Helped |
| use the minimax model instead of LFlows | optimal transport estimation · Dynamical optimal transport: toy 2D distributions | W₂² estimate | Hurt |
| use DFNNs instead of LFlows | optimal transport estimation · Dynamical optimal transport: toy 2D distributions | W₂² estimate | Hurt |
| use methods enforcing the continuity equation instead of the baseline MLP | bird migration density prediction · Bird migration | test density error | Helped |
Surprising findings
- DFNNs and SLDA are roughly comparable to LFlows in density MSE on bird migration, but suffer from high compute and memory costs.
- DFNN velocity computed as flux divided by density is numerically unstable in low-density regions, so DFNN consistency results were omitted.
Limitations
- Like normalizing flows, disconnected modes in the target require matching disconnected modes in the base density, leaving small nonzero density between modes.
- Expressive power is bounded by the bijective layers, each limited in the number of modes it can model.
- Total mass c is a learned hyperparameter and requires a mass penalty; the bird-migration data has no z velocity component.
Experimental setup
Datasets: synthetic Gaussian-mixture fluid flow (introduced) 2D toy OT datasets (moons, swissroll, circles, pinwheel, 8gaussians) European weather radar bird migration data
Benchmarks: synthetic fluid flow 2D and 3D dynamical optimal transport toy 2D datasets bird migration radar density
Metrics: R^2 (explained variance) of density sMAPE consistency test MSE of log1p density W2^2 distance estimate peak VRAM time per epoch
Baselines: PINN DFNN SLDA MLP minimax OT via input convex neural networks discrete OT solver (POT)
Builds on: conditional normalizing flows invertible densenets physics-informed neural networks divergence-free neural networks continuous normalizing flows
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