Localized Dynamic Mode Decomposition with Temporally Adaptive Segmentation

TL;DR

This work introduces Localized DMD (LDMD), a time-domain segmentation framework that applies DMD within short, locally linear temporal windows to improve long-term predictive accuracy for nonlinear dynamical systems. It develops two segmentation strategies: predefined LDMD (P-LDMD) and adaptive LDMD (A-LDMD), the latter using a residual-based error estimator to dynamically adjust stage boundaries and data usage. The authors provide an error-analysis showing upper bounds on local and global truncation errors and validate LDMD on Burgers’, Allen–Cahn, nonlinear Schrödinger, and Maxwell’s equations, where LDMD consistently outperforms standard DMD and several variants in accuracy and efficiency. The results suggest LDMD’s potential as a high-accuracy surrogate model for parametric and complex dynamical systems, with future directions including data-driven error estimators and integration with rigorous residual-DMD frameworks.

Abstract

Dynamic mode decomposition (DMD) is a widely used data-driven algorithm for predicting the future states of dynamical systems. However, its standard formulation often struggles with poor long-term predictive accuracy. To address this limitation, we propose a localized DMD (LDMD) framework that improves prediction performance by integrating DMD's strong linear forecasting capabilities with time-domain segmentation techniques. In this framework, the temporal domain is segmented into multiple subintervals, within which snapshot matrices are constructed and localized predictions are performed. We first present the localized DMD method with predefined segmentation, and then explore an adaptive segmentation strategy to further enhance computational efficiency and prediction robustness. Furthermore, we conduct an error analysis that provides the upper bound of the local and global truncation error for the proposed framework. The effectiveness of LDMD is demonstrated on four benchmark problems-Burgers', Allen-Cahn, nonlinear Schrodinger, and Maxwell's equations. Numerical results show that LDMD significantly enhances long-term predictive accuracy while preserving high computational efficiency.

Figures (30)

• Figure 1: The solution $\mathbf{u}(t,\mathbf{x})$ of the Burgers’ equation: (a) the reference solution, (b) the standard DMD solution, and (c) the A-LDMD solution.
• Figure 2: (a) The $L^2$ relative error of the Burgers' equation for DMD, A-LDMD, mrDMD, and POD-RBF at $\gamma=50\%$. (b) Residual of the A-LDMD solution.
• Figure 3: The $L^2$ relative error across different prediction rates. From left to right, prediction rates $\gamma$ are $40\%,~50\%,~60\%$ respectively.
• Figure 4: The $L^2$ relative error across different residual error thresholds.
• Figure 5: Comparison of the number and location of segments under different residual error thresholds $\epsilon$: (a) $\epsilon=10^{-2}$, (b) $\epsilon=10^{-4}$, and (c) $\epsilon=5\times10^{-5}$. The red dashed lines indicate the locations of the segmentation boundaries.

Theorems & Definitions (26)

• Definition 2.1: The family of Koopman operator
• Definition 2.2: Koopman operator dynamic2016kutz
• Remark 2.1
• Remark 3.1
• Remark 3.2
• Lemma 4.1: Prediction2020Lu
• Lemma 4.2: Prediction2020Lu
• Theorem 4.1
• proof
• Lemma 4.3: Prediction2020Lu