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A Generalizable Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting in Complex Environments
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Proposes Phy-SSM, a state-space model that embeds partially known physics via a knowledge-masked decomposition of the state matrix plus a physics state regularization term, for long-term forecasting on noisy, irregular data.
Problem
Long-term dynamics forecasting (interpolation and extrapolation) in complex environments where data are noisy, irregularly sampled, and only partial physics knowledge is available.
Approach
Decompose partially known dynamics into known and unknown state matrices, enforce known physics via a binary knowledge mask (Hadamard product) inside a continuous-time SSM within a VAE-style encoder/decoder, and add a physics state regularization term aligning latent states with system dynamics.
Key results
- On the drone dataset, Phy-SSM reports extrapolation MAE 2.733e-1 and MSE 1.798e-1, versus 8.426e-1 / 1.733 for S5 (the best baseline by MSE is 2.176e-1 in the ablation only).
- On COVID-19 extrapolation, Phy-SSM achieves MAE 1.998e-1 and MSE 0.692e-1, against 5.212e-1 and 4.560e-1 for S5.
- On nuScenes out-of-domain prediction, Phy-SSM achieves ADE 6.206 and FDE 7.197, the best among compared methods.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Integrate partial physics in the Phy-SSM unit (ablation) | Phy-SSM · Drone extrapolation | MAE: 8.426$\pm$1.351 -> 3.008$\pm$0.053 | Helped |
| Add physics state regularization (ablation) | Phy-SSM · Drone extrapolation | MAE: 3.008$\pm$0.053 -> 2.733$\pm$0.059 | Helped |
| Add physics state regularization (ablation) | Phy-SSM · Drone interpolation | MAE: 0.927$\pm$0.057 -> 1.002$\pm$0.034 | Hurt |
| Use Phy-SSM instead of the strongest listed baseline | Drone forecasting · Drone interpolation and extrapolation | MAE: Interpolation: 1.059$\pm$0.122 -> 1.002$\pm$0.034; extrapolation: 3.456$\pm$0.289 -> 2.733$\pm$0.059 | Helped |
| Use Phy-SSM instead of the strongest listed baseline | COVID-19 forecasting · COVID-19 interpolation and extrapolation | MAE: Interpolation: 0.830$\pm$0.156 -> 0.795$\pm$0.208; extrapolation: 5.212$\pm$0.554 -> 1.998$\pm$0.753 | Helped |
| Use Phy-SSM instead of the strongest listed baseline | Vehicle motion prediction · nuScenes in-domain extrapolation | FDE: 4.954$\pm$0.086 -> 5.100$\pm$0.160 | Hurt |
| Use Phy-SSM instead of the strongest listed baseline | Vehicle motion prediction · nuScenes out-of-domain extrapolation | ADE: 7.050$\pm$0.543 -> 6.206$\pm$0.229 | Helped |
| Use Phy-SSM instead of the strongest listed baseline | Vehicle motion prediction · nuScenes out-of-domain extrapolation | Speed Error: 2.381$\pm$0.254 -> 2.398$\pm$0.532 | Hurt |
Surprising findings
- Removing the physics state regularization improves interpolation but substantially degrades extrapolation, showing a trade-off between fitting observations and generalizing.
Limitations
- Physics knowledge must be partially specified by domain experts and a knowledge mask designed per system.
- Vehicle motion out-of-domain results show high variance on some baselines; in-domain gains over Wayformer on ADE/FDE are small or absent.
- Regularization weight and beta hyperparameters require task-specific tuning.
Experimental setup
Datasets: nuScenes quadrotor drone dataset COVID-19 epidemiology data video pendulum
Benchmarks: nuScenes drone state prediction COVID-19 forecasting
Metrics: MAE MSE ADE FDE speed error acceleration error jerk error
Baselines: Latent ODE (RNN Enc.) Latent ODE (ODE-RNN Enc.) ContiFormer S5 GOKU PI-VAE SDVAE ODE2VAE Wayformer AutoBot G2LTraj
Builds on: S4 S5 HiPPO GOKU ODE2VAE Physics-Integrated VAE
Closest papers by meaning
- From Generalization Analysis to Optimization Designs for State Space Models (ICML 2024 · similarity 0.87)
- Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators (NeurIPS 2026 · similarity 0.86)
- Predicting partially observable dynamical systems via diffusion models with a multiscale inference scheme (NeurIPS 2025 · similarity 0.86)
- Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series (ICLR 2025 · similarity 0.85)
- Physiome-ODE: A Benchmark for Irregularly Sampled Multivariate Time-Series Forecasting Based on Biological ODEs (ICLR 2025 · similarity 0.85)
- ABC: Any-Subset Autoregression via Non-Markovian Diffusion Bridges in Continuous Time and Space (NeurIPS 2026 · similarity 0.85)
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