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P$^2$C$^2$Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics
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Proposes P²C²Net, a physics-encoded coarse-grid correction network that combines a trainable PDE block (RK4 time marching with a learnable 7-parameter symmetric conv filter for derivatives) with an FNO correction block, achieving over 50% lower error than baselines on four PDE datasets with only 3–5 training trajectories.
Problem
Predicting spatiotemporal PDE dynamics on coarse mesh grids with large time steps and only a small amount of training data, where classical numerical schemes diverge and purely data-driven models need rich labeled data.
Approach
Combine a trainable PDE block whose spatial derivatives come from a learnable symmetric convolution filter (sum-rule constrained, 7 parameters, 5x5) applied to a neural-corrected state, integrated with RK4, plus an FNO correction block that restores information lost at low resolution; periodic boundary padding enforces BCs and a spectral Poisson block recovers pressure for Navier-Stokes.
Key results
- On Burgers, P²C²Net reduces RMSE from 0.0967 (best baseline PeRCNN) to 0.0064, a 93.4% promotion, with HCT of 1.4 s vs 0.45 s.
- On NS (Re=1000, 64x64 coarse grid), RMSE drops to 0.3533 from 0.8224 (UNet), a 57.0% improvement, with HCT 7.20 s vs 3.96 s.
- On FN, RMSE improves 77.5% over the best baseline (UNet 0.1730 to 0.0390); on GS, RMSE improves 91.5% (PeRCNN 0.1586 to 0.0135).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use P$^2$C$^2$Net instead of PeRCNN | Burgers · Burgers | RMSE: 0.0967 -> 0.0064 | Helped |
| use P$^2$C$^2$Net instead of PeRCNN | GS · GS | RMSE: 0.1586 -> 0.0135 | Helped |
| use P$^2$C$^2$Net instead of UNet | FN · FN | RMSE: 0.1730 -> 0.0390 | Helped |
| use P$^2$C$^2$Net instead of UNet | NS · NS | RMSE: 0.8224 -> 0.3533 | Helped |
| add symmetric convolution constraints (ablation) | P$^2$C$^2$Net on Burgers · Burgers | RMSE: 0.1450 -> 0.0064 | Helped |
| use learnable symmetric filters instead of finite-difference stencil kernels (ablation) | P$^2$C$^2$Net on Burgers · Burgers | RMSE: NaN -> 0.0064 | Helped |
| add the correction block (ablation) | P$^2$C$^2$Net on Burgers · Burgers | RMSE: 0.1463 -> 0.0064 | Helped |
| use RK4 instead of first-order Euler integration (ablation) | P$^2$C$^2$Net on Burgers · Burgers | RMSE: 0.0322 -> 0.0064 | Helped |
| use a 5×5 filter instead of a 3×3 filter (ablation) | P$^2$C$^2$Net on Burgers · Burgers | RMSE: 0.1274 -> 0.0064 | Helped |
| use a 5×5 filter instead of a 7×7 filter (ablation) | P$^2$C$^2$Net on Burgers · Burgers | RMSE: NaN -> 0.0064 | Helped |
| add 1% noise to training data (ablation) | P$^2$C$^2$Net on Burgers · Burgers | RMSE: 0.0064 -> 0.0092 | Hurt |
| reduce Burgers training data by 20% (ablation) | P$^2$C$^2$Net · Burgers | RMSE: 0.0064 -> 0.0073 | Hurt |
Surprising findings
- Replacing the 5x5 symmetric filter with a 3x3 or 7x7 filter, or using fixed finite-difference stencils, produces NaN or much larger errors in the ablation, indicating sensitivity to kernel design.
- The NN correction block is reported as unnecessary for simpler systems such as Burgers, where the PDE block alone performs well.
Limitations
- Restricted to regular grids with periodic boundary conditions; irregular grids and other BC types are not handled.
- Only 2D problems are studied; extension to 3D is future work.
- Some baseline comparisons use NaN or diverging results, and the paper uses a single mesh configuration per dataset.
Experimental setup
Datasets: 2D Burgers (FD-simulated) Gray-Scott (FD-simulated) FitzHugh-Nagumo (FD-simulated) 2D Navier-Stokes Kolmogorov flow (FV-simulated)
Benchmarks: 2D Burgers Gray-Scott FitzHugh-Nagumo Navier-Stokes (Re=1000)
Metrics: RMSE MAE MNAD HCT
Baselines: FNO UNet PeRCNN DeepONet Learned Interpolation (LI)
Builds on: FNO PeRCNN learned interpolation RK4 physics-encoded learning
Closest papers by meaning
- PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.92)
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