Entropic Regression DMD (ERDMD) Discovers Informative Sparse and Nonuniformly Time Delayed Models

TL;DR

ERDMD addresses the challenge of learning accurate, parsimonious multi-step DMD models from chaotic data by automatically discovering nonuniform time delays using entropic regression. It merges Higher-Order DMD with causation-entropy-based model discovery to select lag sets that maximize information flow, producing sparse Koopman-like operators for time stepping. In Lorenz-63 experiments, ERDMD yields compact lag structures and interpretable spectra, with reconstruction approaching but not always matching full HODMD and forecasting demonstrating robustness to overfitting. Overall, the method offers a noise-tolerant, information-theoretic framework for data-driven time-series modeling with potential to improve diagnostic insight into multiscale dynamics.

Abstract

In this work, we present a method which determines optimal multi-step dynamic mode decomposition (DMD) models via entropic regression, which is a nonlinear information flow detection algorithm. Motivated by the higher-order DMD (HODMD) method of \cite{clainche}, and the entropic regression (ER) technique for network detection and model construction found in \cite{bollt, bollt2}, we develop a method that we call ERDMD that produces high fidelity time-delay DMD models that allow for nonuniform time space, and the time spacing is discovered by consider most informativity based on ER. These models are shown to be highly efficient and robust. We test our method over several data sets generated by chaotic attractors and show that we are able to build excellent reconstructions using relatively minimal models. We likewise are able to better identify multiscale features via our models which enhances the utility of dynamic mode decomposition.

Figures (13)

• Figure 1: The BUILD stage of the ERDMD algorithm. Here, we start from the given model ${\bf M}_{c}$ which represents the choice of lags $(1, \cdots, l_{k})$ with corresponding matrices $({\bf K}_{1}, \cdots, {\bf K}_{l_{k}})$, and we then find a proposed model ${\bf M}^{(j)}_{t}$ which maximizes $I\left(\left.{\bf y}^{+}, {\bf M}^{(j)}_{t}\right|{\bf M}_{c}\right)$, or the information gain in using the proposed model relative to the given model to anticipate the next time steps represented by ${\bf y}^{(+)}$.
• Figure 2: Direct comparison of ERDMD and all lags HODMD model against the true trajectory for the Lorenz-63 system (a), and error across dimensions for the ERDMD and all lags HODMD method (b). The black line indicates the ERDMD result while the red indicates the all lags HODMD result. The solid vertical bar indicates the maximum lag choice of $d=150$, while the dashed line indicates the end of the reconstruction interval and the beginning of the forecasting regime. The ERDMD algorithm converges to $l_{c}=\left\{1,149\right\}$.
• Figure 3: Comparison of the HODMD lagged matrix norms (black dots) and the ERDMD model (red dots) for the Lorenz-63 system with $d=150$.
• Figure 4: Spectrum of corresponding ERDMD Koopman operator for Lorenz-63 with $d=150$ on the left side, with a detail comparison to the roots of $\tilde{p}_{in, a}(z)$ (blue crosses) on the right side near (1,0). The ERDMD algorithm converges to $l_{c}=\left\{1,149\right\}$. The solid/red line is the unit circle, provided for reference.
• Figure 5: Comparison of ERDMD and all lags HODMD model against true trajectory for Lorenz-63 system. The ERDMD reconstruction is in solid black in the figure. The vertical bar indicates the maximum lag choice of $d=100$. The ERDMD algorithm converges to $l_{c}=\left\{1,15, 26, 35, 45, 48, 68, 73, 97,99\right\}$.