Online Physics-Informed Dynamic Mode Decomposition: Theory and Applications

TL;DR

OPIDMD addresses the limitations of traditional DMD by embedding physical constraints into a convex online optimization framework, yielding a time-varying operator $A_k$ learned in real time. The authors connect physics-informed formulations to convex relaxations, leveraging proximal gradient descent to enforce constraints while maintaining convergence guarantees. A Bayesian interpretation and MAP regularization unify probabilistic reasoning with physics-based priors, while online proximal methods enable efficient, real-time updates across multiple physical scenarios. Empirical results on advection, cylinder flow, and the Lorenz system show improved short-term prediction accuracy and noise robustness, with the Lorenz example achieving $R^2 = 0.991$, highlighting OPIDMD’s potential for real-time simulation and control of complex dynamical systems.

Abstract

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.

Figures (26)

• Figure 1: Framework of the Online Physics-Informed DMD. This diagram illustrates the comprehensive framework of this paper, emphasizing the physics-informed formulation of DMD, which can be interpreted in two ways. The left branch focuses on probabilistic aspects, such as Bayesian interpretation, allowing us to perform bias-variance analysis in DMD. The right branch emphasizes physical principles; however, some physical constraints may lead to non-convex optimization problems, necessitating a transition from non-convex to convex strategies. By leveraging convex optimization and duality theory, both branches ultimately converge to the same solution. Finally, the model can be solved using online proximal gradient descent for piDMD.
• Figure 2: A schematic representation of the OPIDMD algorithm process.
• Figure 3: Initial Conditions of the Two-Dimensional Advection Equation
• Figure 4: Comparison of DMD Matrices
• Figure 5: Eigenvalue Distributions