Multi-trajectory Dynamic Mode Decomposition

Abstract

We propose a new interpretation of the parameter optimization in dynamic mode decomposition (DMD), and show a rigorous application in multi-trajectory modeling. The proposed method, multitrajectory DMD (MTDMD) is a numerical method with which we can model a dynamical system using multiple trajectories, or sets of consecutive snapshots. Here, a trajectory corresponds to an experiment, so the proposed method accepts multiple experimental results. This is in contrast to any of the existing DMD-based methods, that accepts only one set of consecutive snapshots. The incorporation of the multi-trajectory modeling in DMD is beneficial because we can suppress the
undesirable effects from the observation noises. Also, we can perform multiple experiments with different experimental conditions to model a complex system with simple experimental set-ups. We test the proposed method against the numerically generated synthetic data, and show that the
MTDMD achieves not only good reconstruction results but also numerically fast multi-trajectory modeling for multi-dimensional time-series data. The proposed method is quite flexible, and we suggest that we can also implement L2 regularization and equality constraints on the model parameters. The proposed method is expected to serve as an effective method to model large systems using noisy time-series data without elaborate design
of experiment.