Research map / Probabilistic and scientific modeling / Differential equations and PDEs
MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation
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Proposes MultiPDENet, a PDE-embedded network that combines a learnable symmetric-filter PDE block integrated with RK4 at a fine time scale and a UNet macro-step correction to accelerate coarse-grid flow simulation from little data.
Problem
Accelerating numerical PDE simulation (e.g., turbulent flows) on coarse grids with limited training data, while retaining generalization, interpretability, and long-term stability against error accumulation.
Approach
Embed a learnable PDE residual (symmetric finite-difference-like convolution filters with few parameters) inside an RK4 integrator at micro time steps, correct the intermediate states with FNO and a Poisson spectral pressure solver, and add a UNet macro-step correction to limit temporal error accumulation.
Key results
- On Burgers, MultiPDENet reduces RMSE to 0.0057 vs 0.0967 for the best baseline (PeRCNN), a 94.1% improvement.
- On NSE Kolmogorov flow (Re=1000), MultiPDENet achieves RMSE 0.1379 and HCT 8.36 s, versus 0.8224 RMSE for UNet, with 83.2% improvement.
- Reports at least 5x speedup over GPU-accelerated DNS (JAX-CFD) at correlation >= 0.8 on a single A100.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use MultiPDENet instead of the strongest reported baseline, PhyFNO | KdV · KdV | RMSE: 0.4120 -> 0.1536 | Helped |
| use MultiPDENet instead of the strongest reported baseline, PeRCNN | Burgers · Burgers | RMSE: 0.0967 -> 0.0057 | Helped |
| use MultiPDENet instead of the strongest reported baseline, PeRCNN | GS · GS | RMSE: 0.1763 -> 0.0573 | Helped |
| use MultiPDENet instead of the strongest reported baseline, UNet | NSE · NSE | RMSE: 0.8224 -> 0.1379 | Helped |
| add the Poisson Block (ablation) | MultiPDENet · KF | RMSE: 0.1601 -> 0.1379 | Helped |
| add the filter structure constraint (ablation) | MultiPDENet · KF | RMSE: 0.2432 -> 0.1379 | Helped |
| add the MaNN Block to the Physics Block-only model (ablation) | MultiPDENet · KF | RMSE: 0.2632 -> 0.1379 | Helped |
| use the symmetric filter instead of FD convolution (ablation) | MultiPDENet · KF | RMSE: 0.2503 -> 0.1379 | Helped |
| add the Correction Block (ablation) | MultiPDENet · KF | RMSE: 0.2958 -> 0.1379 | Helped |
| add the MiNN Block (ablation) | MultiPDENet · KF | RMSE: 0.4338 -> 0.1379 | Helped |
| add the MaNN Block (ablation) | MultiPDENet · KF | HCT (s): 1.4193 -> 8.3566 | Helped |
| add the Physics Block (ablation) | MultiPDENet · KF | RMSE: 1.2023 -> 0.1379 | Helped |
Surprising findings
- DeepONet diverges (NaN) on KdV and UNet/FNO produce NaN on some datasets, showing baseline instability on coarse data.
- Physics Block alone (Model C) without the macro-step NN is worse than the full model but still gives reasonable predictions, while removing the macro NN (Model G) causes NaN divergence.
Limitations
- Only handles regular grids due to convolution operations.
- Only tested on 1D and 2D problems; 3D not yet addressed.
- Training uses very few trajectories, and baselines' weaker performance may depend on the coarse-data setup.
Experimental setup
Datasets: KdV (spectral DNS, downsampled) Burgers (FD, downsampled) Gray-Scott (FD, downsampled) Kolmogorov flow NSE (JAX-CFD-style FV, downsampled)
Benchmarks: KdV Burgers Gray-Scott Kolmogorov flow NSE
Metrics: RMSE MAE MNAD HCT
Baselines: FNO UNet TSM LI DeepONet PeRCNN PhyFNO
Builds on: PeRCNN Fourier Neural Operator RK4 integration PINN
Closest papers by meaning
- P$^2$C$^2$Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics (NeurIPS 2024 · similarity 0.90)
- SineNet: Learning Temporal Dynamics in Time-Dependent Partial Differential Equations (ICLR 2024 · similarity 0.90)
- CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs (NeurIPS 2025 · similarity 0.90)
- Text2PDE: Latent Diffusion Models for Accessible Physics Simulation (ICLR 2025 · similarity 0.89)
- Mechanistic PDE Networks for Discovery of Governing Equations (ICML 2025 · similarity 0.89)
- PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.89)
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