在线物理约束动态模态分解，提升复杂系统实时建模精度与稳定性。
Online Physics-Informed Dynamic Mode Decomposition: Theory and Applications
- 将DMD融入凸优化框架，结合物理规律约束，实现在线更新。
- 在含噪洛伦兹系统上达到0.991的R²预测精度，优于现有方法。
- 适合需要实时仿真与控制的工程系统，如流体、机器人等。
动态模态分解（DMD）因其对复杂动力系统的分析与建模能力而受到广泛关注。然而，其在计算效率、抗噪性以及遵守物理定律方面存在挑战，影响实际性能。为此，本文提出在线物理信息型动态模态分解（OPIDMD），将DMD重构为凸优化问题，确保收敛至唯一全局最优解，并显著提升在线场景下的建模效率与精度。基于贝叶斯DMD框架，我们给出物理约束下DMD线性算子的概率解释，分析物理先验的影响。进一步设计了在线近端梯度下降算法，针对不同物理约束制定具体求解流程，支持多种场景下的实时求解。相比经典方法（如Exact DMD、Online DMD、piDMD），OPIDMD在短期预测中表现最优，例如在含噪洛伦兹系统上取得0.991的R²值。该方法采用时变线性算子，为复杂动力系统的实时仿真与控制提供有效解决方案。
原文摘要 · Abstract (English)
Dynamic Mode Decomposition (DMD) has received increasing research attention due to its capability to analyze and model complex dynamical systems. However, it faces challenges in computational efficiency, noise sensitivity, and difficulty adhering to physical laws, which negatively affect its performance. Addressing these issues, we present Online Physics-informed DMD (OPIDMD), a novel adaptation of DMD into a convex optimization framework. This approach not only ensures convergence to a unique global optimum, but also enhances the efficiency and accuracy of modeling dynamical systems in an online setting. Leveraging the Bayesian DMD framework, we propose a probabilistic interpretation of Physics-informed DMD (piDMD), examining the impact of physical constraints on the DMD linear operator. Further, we implement online proximal gradient descent and formulate specific algorithms to tackle problems with different physical constraints, enabling real-time solutions across various scenarios. Compared with existing algorithms such as Exact DMD, Online DMD, and piDMD, OPIDMD achieves the best prediction performance in short-term forecasting, e.g. an $R^2$ value of 0.991 for noisy Lorenz system. The proposed method employs a time-varying linear operator, offering a promising solution for the real-time simulation and control of complex dynamical systems.
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