A Generalizable Physics-Enhanced State Space Model for Long-Term Dynamics Forecasting in Complex Environments

Yuchen Wang, Hongjue Zhao, Haohong Lin, Enze Xu, Lifang He, Huajie Shao

TL;DR: We propose Phy-SSM, a general-purpose framework that integrates partial physics knowledge into state space models (SSMs) for long-term dynamics forecasting.

Abstract: This work aims to address the problem of long-term dynamic forecasting in complex environments where data are noisy and irregularly sampled. While recent studies have introduced some methods to improve prediction performance, these approaches still face a significant challenge in handling long-term extrapolation tasks under such complex scenarios. To overcome this challenge, we propose Phy-SSM, a general-purpose framework that integrates partial physics knowledge into state space models (SSMs) for long-term dynamics forecasting in complex environments. Our motivation is that SSMs can effectively capture long-range dependencies in sequential data and model continuous dynamical systems, while the incorporation of physics knowledge improves generalization ability. The key challenge lies in how to seamlessly incorporate partially known physics into SSMs. To achieve this, we decompose partially known system dynamics into known and unknown state matrices, which are integrated into a Phy-SSM unit. To further enhance long-term prediction performance, we introduce a physics state regularization term to make the estimated latent states align with system dynamics. Besides, we theoretically analyze the uniqueness of the solutions for our method. Extensive experiments on three real-world applications, including vehicle motion prediction, drone state prediction, and COVID-19 epidemiology forecasting, demonstrate the superior performance of Phy-SSM over the baselines in both long-term interpolation and extrapolation tasks. The source code will be publicly available upon publication.

Lay Summary:

Predicting how systems evolve over time — such as the motion of cars and drones, or the spread of disease — is challenging in real-world settings, especially when data is noisy and our understanding of the underlying physics is incomplete. We wondered: can we still make accurate predictions using only partial knowledge of the physical system?

To tackle this, we developed Phy-SSM — a method that seamlessly combines partially known physics with a deep state space model, a type of AI that excels at learning from time series data. Our key innovation is to explicitly separate the known and unknown parts of the physical system, then merge them in a unified framework that helps the model learn the true system dynamics. Surprisingly, we found that even incomplete physics knowledge can significantly improve the model’s ability to generalize for long-term predictions on unseen data.

Phy-SSM allows users to easily incorporate partial physics knowledge into different systems to make more accurate forecasts. Our research highlights the potential of combining traditional physics with modern AI to tackle real-world prediction challenges.

Link To Code: https://github.com/511205787/Phy_SSM-ICML2025

Primary Area: Deep Learning->Sequential Models, Time series

Keywords: Physics-enhanced Machine Learning, State Space Model, Long-term Dynamics Forecasting, Dynamical Systems

Submission Number: 8797