Meta-Learning for Koopman Spectral Analysis with Short Time-series

TL;DR

This paper introduces a meta-learning approach to estimate Koopman embedding functions from short time-series, enabling spectral analysis of nonlinear systems when data is limited, by leveraging knowledge from related time-series.

Contribution

The proposed method allows effective Koopman spectral analysis from short time-series using meta-learning, which was not feasible with previous neural network-based methods requiring long data sequences.

Findings

Outperforms existing methods in eigenvalue estimation

Achieves better future prediction accuracy

Effective with limited short time-series data

Abstract

Koopman spectral analysis has attracted attention for nonlinear dynamical systems since we can analyze nonlinear dynamics with a linear regime by embedding data into a Koopman space by a nonlinear function. For the analysis, we need to find appropriate embedding functions. Although several neural network-based methods have been proposed for learning embedding functions, existing methods require long time-series for training neural networks. This limitation prohibits performing Koopman spectral analysis in applications where only short time-series are available. In this paper, we propose a meta-learning method for estimating embedding functions from unseen short time-series by exploiting knowledge learned from related but different time-series. With the proposed method, a representation of a given short time-series is obtained by a bidirectional LSTM for extracting its properties. The embedding function of the short time-series is modeled by a neural network that depends on the time-series representation. By sharing the LSTM and neural networks across multiple time-series, we can learn common knowledge from different time-series while modeling time-series-specific embedding functions with the time-series representation. Our model is trained such that the expected test prediction error is minimized with the episodic training framework. We experimentally demonstrate that the proposed method achieves better performance in terms of eigenvalue estimation and future prediction than existing methods.