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Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws
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Introduces clawNOs, neural operators whose outputs are divergence-free by construction (curl of a skew-symmetric matrix field via a fixed numerical differentiation layer), enforcing mass conservation automatically and improving small-data accuracy across Navier-Stokes, shallow water, atmospheric and material deformation tasks.
Problem
Neural operators learn physical laws from data and only approximately satisfy conservation laws (e.g., mass/continuity) due to finite and noisy data, which hurts accuracy and physical consistency especially in small-data regimes.
Approach
Parameterize the output as the divergence of a skew-symmetric matrix-valued function (from differential forms, d-star-d = 0), so the continuity equation holds automatically; the divergence is computed by a pre-calculated numerical differentiation layer (Fourier spectral differentiation on periodic uniform grids, or meshfree weighted finite differences on non-uniform grids) appended to existing FNO
Key results
- On Navier-Stokes (T_in=10, T=20) with 10 training samples, clawGFNO achieves 12.10% relative L2 error vs 29.56% for GFNO-p4 and 22.57% for FNO; clawFNO reaches 12.68%.
- On atmospheric modeling, clawFNO achieves 6.64% test error vs 12.54% for FNO, about 47% better than the best non-claw baseline.
- On radial dam break with 2 training samples, clawFNO achieves 21.24% vs 27.31% for the best non-claw baseline (GFNO-p4).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add the divergence-free conservation-law architecture to GFNO (ablation) | GFNO · Navier–Stokes, T_in=10, T=20; 10, 100, and 1000 training samples | relative L2 error: 29.56% ± 3.32% -> 12.10% ± 1.08%; 7.59% ± 0.34% -> 4.76% ± 0.24%; 3.11% ± 0.93% -> 1.21% ± 0.09% | Helped |
| Add the divergence-free conservation-law architecture to FNO (ablation) | FNO · Navier–Stokes, T_in=10, T=20; 10, 100, and 1000 training samples | relative L2 error: 22.57% ± 1.87% -> 12.68% ± 0.85%; 5.68% ± 0.41% -> 5.17% ± 0.27%; 1.47% ± 0.18% -> 1.46% ± 0.11% | Helped |
| Use equivariant GFNO rather than FNO as the clawNO base | clawNO · Navier–Stokes, T_in=10, T=20 | relative L2 error: 12.68% ± 0.85% -> 12.10% ± 1.08%; 5.17% ± 0.27% -> 4.76% ± 0.24%; 1.46% ± 0.11% -> 1.21% ± 0.09% | Helped |
| Use clawGFNO instead of the strongest non-claw baseline | Navier–Stokes models · Navier–Stokes, both rollout settings | relative L2 error: 40.3%, 16.2% and 17.6% in case 1; 39.9%, 26.8% and 9.1% in case 2 | Helped |
| Increase GFNO latent dimension from 11 to 20 (ablation) | GFNO · Navier–Stokes, T_in=10, T=20, 1000 training samples | relative L2 error: 1.19% ± 0.16% | Helped |
| Add the divergence-free conservation-law architecture to FNO (ablation) | FNO · Radial dam break, 2 and 10 training samples | relative L2 error: 29.88% ± 0.17% -> 21.24% ± 1.17%; 4.04% ± 0.10% -> 2.69% ± 0.04% | Helped |
| Add the divergence-free conservation-law architecture to FNO (ablation) | FNO · Radial dam break, 100 training samples | relative L2 error: 1.88% ± 0.05% -> 1.84% ± 0.01% | Helped |
| Use clawNO rather than the best non-claw baseline | Radial dam break models · Radial dam break, 2 and 10 training samples | relative L2 error: 22.2% in small data regime; 29.2% in medium data regime | Helped |
| Add the divergence-free conservation-law architecture to FNO (ablation) | FNO · Atmospheric modeling | relative L2 error: 12.54% ± 0.26% -> 6.64% ± 0.23% | Helped |
| Add the divergence-free conservation-law architecture to INO (ablation) | INO · Material deformation, 2, 20, and 100 training samples | relative L2 error: 16.75% ± 1.24% -> 14.12% ± 0.23%; 7.69% ± 0.51% -> 5.41% ± 0.67%; 2.40% ± 0.53% -> 1.00% ± 0.02% | Helped |
| Use clawINO rather than the best non-claw baseline | Material deformation models · Material deformation, 2, 20, and 100 training samples | relative L2 error: 15.7%, 26.4%, and 23.1% | Helped |
| Use clawNOs rather than unconstrained neural operators | Neural operators · Navier–Stokes test set, 1000 training samples | L2-norm divergence: 1.2e-3; 3.8e-4 vs 6.6e-2; 4.7e-2 | Helped |
Surprising findings
- LNN fails to learn the Navier-Stokes solution with test error above 20% across all data regimes, attributed to temporal integration errors on temporally sparse data.
- Increasing GFNO latent width to match FNO yields 1.19% error in large-data Navier-Stokes, suggesting parameter-matched comparisons alter baseline rankings.
Limitations
- Divergence is not exactly zero due to numerical differentiation errors; error estimates do not apply to solutions with shocks or discontinuities.
- Mixed or non-divergence-free physics requires extra handling; the method is limited to conservation laws expressible as divergence-free fields.
- Experiments use relatively small datasets (1,000 samples max) and rely on simulated rather than experimental data.
Experimental setup
Datasets: incompressible Navier-Stokes simulations (pseudo-spectral Crank-Nicolson solver) radial dam break simulations SpeedyWeather.jl atmospheric simulations Mooney-Rivlin material deformation simulations
Benchmarks: incompressible Navier-Stokes radial dam break atmospheric shallow water (SpeedyWeather.jl) Mooney-Rivlin material deformation
Metrics: relative L2 error L2 norm of divergence trainable parameter count runtime GPU memory
Baselines: FNO G-FNO UNet LSM KNO UNO BOON LNN GNO INO MP-PDE MeshGraphNets
Builds on: Fourier neural operator graph neural operator invariant neural operator group-equivariant FNO physics-informed neural networks
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