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Breaking the Discretization Barrier of Continuous Physics Simulation Learning
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Proposes CoPS, a purely data-driven framework that combines multiplicative filter network encoding, customized grid message passing, multi-scale graph ODEs and a Markov-style neural auto-correction module to forecast physical dynamics continuously in space and time from sparse partial observations.
Problem
Modeling complex time-evolving physical dynamics from sparse, unstructured partial observations while allowing predictions at arbitrary spatial locations and continuous times, without fixed spatial or temporal discretization.
Approach
Encodes coordinates with a multiplicative Gabor filter network, maps irregular observations onto customized uniform grids via message passing, evolves latent states with a multi-scale graph ODE, and applies a convolutional Markov-style auto-correction at discrete steps to bound error accumulation.
Key results
- On Navier-Stokes with 25% observations, CoPS reaches In-s MSE 2.964E-03 versus ContiPDE 5.456E-03 and DINo 1.074E-02.
- On Navier-Stokes with 50% subsampling, CoPS achieves In-t MSE 2.832E-03 versus ContiPDE 5.054E-03 and DINo 7.651E-03.
- On WeatherBench with 75% observations, CoPS reports In-s MSE 3.585E-03 and Ext-s 8.472E-03, lower than baselines.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CoPS instead of ContiPDE | Spatial forecasting from partial observations · Navier-Stokes, Kuroshio, Prometheus, and WeatherBench; 25% observations, In-s | MSE: 5.456E-03 -> 2.964E-03; 2.352E-03 -> 1.285E-03; 8.462E-03 -> 5.176E-03; 1.542E-02 -> 8.766E-03 | Helped |
| Use CoPS instead of ContiPDE | Continuous-time forecasting · Navier-Stokes, 50% observations, In-t / Ext-t / Con-t | MSE: 5.054E-03 -> 2.832E-03; 8.342E-03 -> 5.764E-03; 6.447E-03 -> 4.135E-03 | Helped |
| Add MFN rather than replacing it with an MLP (ablation) | CoPS · Kuroshio, In-s / In-t | MSE: 1.027E-03 -> 7.253E-04 | Helped |
| Add MFN rather than replacing it with an MLP (ablation) | CoPS · Kuroshio, Ext-s / In-t | MSE: 1.946E-03 -> 1.577E-03 | Helped |
| Add MFN rather than replacing it with an MLP (ablation) | CoPS · Kuroshio, In-s / Ext-t | MSE: 2.458E-03 -> 2.161E-03 | Helped |
| Add MFN rather than replacing it with an MLP (ablation) | CoPS · Kuroshio, Ext-s / Ext-t | MSE: 3.149E-03 -> 2.938E-03 | Helped |
| Add multi-scale Graph ODE (ablation) | CoPS · Kuroshio, In-s / In-t | MSE: 1.594E-03 -> 7.253E-04 | Helped |
| Add multi-scale Graph ODE (ablation) | CoPS · Kuroshio, Ext-s / In-t | MSE: 2.357E-03 -> 1.577E-03 | Helped |
| Add multi-scale Graph ODE (ablation) | CoPS · Kuroshio, In-s / Ext-t | MSE: 3.053E-03 -> 2.161E-03 | Helped |
| Add multi-scale Graph ODE (ablation) | CoPS · Kuroshio, Ext-s / Ext-t | MSE: 4.175E-03 -> 2.938E-03 | Helped |
| Add neural auto-correction (ablation) | CoPS · Kuroshio, In-s / In-t | MSE: 1.428E-03 -> 7.253E-04 | Helped |
| Add neural auto-correction (ablation) | CoPS · Kuroshio, Ext-s / In-t | MSE: 2.216E-03 -> 1.577E-03 | Helped |
Limitations
- Theoretical error bound relies on assumed Lipschitz and contraction conditions that may not hold in practice.
- Reported results are single MSE numbers in tables; variance details are not clearly provided in the main text.
Experimental setup
Datasets: Navier-Stokes Rayleigh-Bénard Convection Prometheus WeatherBench Kuroshio
Benchmarks: Navier-Stokes Rayleigh-Bénard Convection Prometheus WeatherBench Kuroshio
Metrics: MSE
Baselines: MAgNet DINo ContiPDE
Builds on: multiplicative filter network Neural ODE GraphCast MeshGraphNet
Closest papers by meaning
- Space and time continuous physics simulation from partial observations (ICLR 2024 · similarity 0.91)
- PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems (ICLR 2025 · similarity 0.87)
- ODEWorld: A Continuous Predictive Architecture via Physical-Time Flow (NeurIPS 2026 · similarity 0.86)
- Predicting partially observable dynamical systems via diffusion models with a multiscale inference scheme (NeurIPS 2025 · similarity 0.86)
- Space-Time Continuous PDE Forecasting using Equivariant Neural Fields (NeurIPS 2024 · similarity 0.86)
- Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solvers (NeurIPS 2025 · similarity 0.86)
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