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Symmetric Basis Convolutions for Learning Lagrangian Fluid Mechanics
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Proposes SFBC, a continuous convolution using separable symmetric Fourier basis functions (no window function) for learning Lagrangian fluid simulations, evaluated on 1D compressible, 2D weakly compressible, and 2D incompressible SPH test cases.
Problem
Learning Lagrangian (particle-based) fluid simulations with continuous convolutions requires inductive biases such as smoothness, symmetry, and window functions, and existing basis choices (linear interpolation, splines, DMCF antisymmetric bases) lack symmetry or smoothness guarantees.
Approach
A general formulation of continuous convolutions as separable basis-function expansions that subsumes existing methods; using a smooth Fourier series with both even and odd terms as the basis enables symmetric, smooth filters so that window functions are no longer needed.
Key results
- SFBC reduces the lowest L2 error on the symmetric kernel toy task by a factor of 5217 compared to LinCConv.
- In the weakly compressible 2D test, the Fourier basis without window outperforms LinCConv by a factor of 2.6 in divergence error.
- CConv-based SFBC uses about 30K parameters versus 229K-395K for MLSConv, GNS, and MP-PDE while matching or beating their accuracy in 1D.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a symmetric, smooth Fourier basis instead of LinCConv | 1D toy kernel interpolation · 1D toy kernel interpolation | L2 error: 5217 | Helped |
| Use an odd-symmetry Fourier series for the kernel gradient | 1D toy kernel gradient · 1D toy kernel gradient | L2 error: a factor of 7.6 relative to DMCF and 853 relative to LinCConv | Helped |
| Use SFBC instead of LinCConv | 3D toy symmetric task · 3D toy symmetric task | L2 error: by a factor of two on average | Helped |
| Use an antisymmetric odd Fourier basis instead of SFBC | 3D toy symmetric task · 3D toy symmetric task | L2 error: three times worse than SFBC | Hurt |
| Increase features per layer from 4 to 32 (ablation) | Compressible 1D SPH network with 3 message-passing steps · Compressible 1D SPH | test error: by an order of magnitude | Helped |
| Increase message-passing steps from 3 to 6 (ablation) | Compressible 1D SPH network · Compressible 1D SPH | network performance: ±10% | No clear effect |
| Use SFBC instead of LinCConv | Weakly-compressible 2D SPH · Weakly-compressible 2D SPH, 64-step inference | divergence error: by a factor of 2.6 | Helped |
| Increase random seeds from 4 to 32 (ablation) | SFBC · Weakly-compressible 2D SPH, 64-step inference | divergence error: 1.29 to 1.30 | No clear effect |
| Add a Müller window to the Fourier basis (ablation) | Fourier basis on weakly-compressible 2D SPH · Weakly-compressible 2D SPH | divergence error: a 2 times increase | Hurt |
| Change the LinCConv window function (ablation) | LinCConv · Weakly-compressible 2D SPH | density error: by 2.1 times | Helped |
| Use volume-preserving or polar coordinate mapping instead of identity mapping (ablation) | SFBC · Weakly-compressible 2D SPH | divergence error: increased by up to 15% | Hurt |
| Replace the first harmonic cosine term with a second harmonic sine term (ablation) | Fourier basis without a window · Weakly-compressible 2D SPH | divergence error: by a factor of 1.4 | Helped |
Surprising findings
- Fourier-based convolutions with no window function outperform networks that use prior window-function inductive biases.
- Antisymmetric-only Fourier bases performed poorly on the non-antisymmetric density-like task, while the combined even/odd basis performed best.
Limitations
- Only networks up to about 200K parameters were considered.
- Only fluid/SPH test cases evaluated; generalization beyond fluid mechanics not shown.
Experimental setup
Datasets: custom SPH simulation datasets (1D compressible, 2D weakly compressible, 2D incompressible)
Benchmarks: toy kernel function and kernel gradient learning 1D compressible SPH 2D weakly compressible SPH 2D incompressible SPH
Metrics: L2 error divergence error density error velocity error power spectral density error point-wise particle distance
Baselines: LinCConv DMCF SplineConv GNS MP-PDE MLSConv PointNet Chebyshev basis
Builds on: LinCConv DMCF SplineConv PointNet MP-PDE
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