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SineNet: Learning Temporal Dynamics in Time-Dependent Partial Differential Equations
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Proposes SineNet, a stack of sequentially connected U-Net "waves" that evolves latent PDE features in small steps to reduce temporal misalignment in skip connections, outperforming U-Nets and FNO variants on Navier-Stokes and shallow water benchmarks.
Problem
Learning time-dependent PDE solution operators with U-Nets suffers from misalignment between skip-connected features and upsampled features, because latent evolution across layers makes features temporally out of sync when advection is significant.
Approach
Partition latent temporal evolution across K sequential U-Net blocks (waves) with wave residuals, so each wave advances the latent state by a small interval and skip connections misalign less; additionally add block residuals that disentangle parallel (direct-from-input) and sequential multi-scale processing paths.
Key results
- SineNet-16 achieves rollout error 17.63% on INS vs 19.29% for the best baseline Dil-ResNet, and 2.04% on CNS vs 2.91% for F-FNO.
- SineNet-8 with ~35.5M parameters reaches SWE rollout error 1.78% vs 2.46% for F-FNO and 3.02% for U-Net-Mod.
- Error decreases monotonically as number of waves K increases from 2 to 16 at fixed parameter budget, with gains plateauing around K=16.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use 8 waves instead of a deeper single U-Net at a similar depth (ablation) | SineNet · INS | rollout error (%): 19.58 -> 19.25 | Helped |
| use 8 waves instead of a deeper single U-Net at a similar depth (ablation) | SineNet · CNS | rollout error (%): 2.76 -> 2.10 | Helped |
| use 8 waves instead of a deeper single U-Net at a similar depth (ablation) | SineNet · SWE | rollout error (%): 2.67 -> 1.78 | Helped |
| disentangle the parallel and sequential processing branches (ablation) | SineNet-8 · INS | rollout error (%): 19.46 -> 19.25 | Helped |
| disentangle the parallel and sequential processing branches (ablation) | SineNet-8 · CNS | rollout error (%): 2.39 -> 2.10 | Helped |
| disentangle the parallel and sequential processing branches (ablation) | SineNet-8 · SWE | rollout error (%): 1.97 -> 1.78 | Helped |
| use circular rather than zero padding (ablation) | SineNet-8 · SWE | 1-step error (%): 1.50 -> 1.02 | Helped |
| use circular rather than zero padding (ablation) | SineNet-8 · SWE | rollout error (%): 4.19 -> 1.78 | Helped |
| increase the number of waves while keeping the parameter budget roughly constant (ablation) | SineNet-K · INS, CNS, and SWE | test error | Helped |
Surprising findings
- Using zero padding instead of circular padding on periodic SWE raises SineNet-8 rollout error from 1.78% to 4.19%.
Limitations
- Increasing number of waves increases inference time and training memory due to added depth.
- CNN-based design cannot handle irregular grids or arbitrary time intervals without extension; out-of-distribution generalization is limited.
- Mixed and non-null boundary conditions are not handled; only zero padding for Dirichlet/Neumann and circular for periodic.
Experimental setup
Datasets: Incompressible Navier-Stokes (INS) Compressible Navier-Stokes (CNS) Shallow Water Equations (SWE) Conditional INS
Benchmarks: INS CNS SWE Conditional INS
Metrics: scaled L2 1-step error scaled L2 rollout error
Baselines: F-FNO Dil-ResNet U-Net-128 U-Net-Mod Deeper U-Net-8 SineNet-8-Entangled
Builds on: U-Net Fourier Neural Operator Neural ODE PDE-Refiner
Closest papers by meaning
- MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation (ICML 2025 · similarity 0.90)
- RandNet-Parareal: a time-parallel PDE solver using Random Neural Networks (NeurIPS 2024 · similarity 0.87)
- CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs (NeurIPS 2025 · similarity 0.87)
- Vectorized Conditional Neural Fields: A Framework for Solving Time-dependent Parametric Partial Differential Equations (ICML 2024 · similarity 0.87)
- Breakeven complexity: A new perspective on neural partial differential equation solvers (NeurIPS 2026 · similarity 0.87)
- Mechanistic PDE Networks for Discovery of Governing Equations (ICML 2025 · similarity 0.86)
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