On Few Shot Learning of Dynamical Systems: A Koopman Operator Theoretic Approach

TL;DR

This paper introduces a new method for learning the Koopman operator from limited, sparsely sampled data by augmenting it with artificial noisy points and applying robust optimization, improving modeling accuracy.

Contribution

The paper presents a novel algorithm that enhances Koopman operator learning from small datasets by data augmentation and robust optimization techniques.

Findings

Effective on linear, nonlinear, and PDE systems

Improves accuracy with sparse data

Demonstrated on three different dynamical systems

Abstract

In this paper, we propose a novel algorithm for learning the Koopman operator of a dynamical system from a \textit{small} amount of training data. In many applications of data-driven modeling, e.g. biological network modeling, cybersecurity, modeling the Internet of Things, or smart grid monitoring, it is impossible to obtain regularly sampled time-series data with a sufficiently high sampling frequency. In such situations the existing Dynamic Mode Decomposition (DMD) or Extended Dynamic Mode Decomposition (EDMD) algorithms for Koopman operator computation often leads to a low fidelity approximate Koopman operator. To this end, this paper proposes an algorithm which can compute the Koopman operator efficiently when the training data-set is sparsely sampled across time. In particular, the proposed algorithm enriches the small training data-set by appending artificial data points, which are treated as noisy observations. The larger, albeit noisy data-set is then used to compute the Koopman operator, using techniques from Robust Optimization. The efficacy of the proposed algorithm is also demonstrated on three different dynamical systems, namely a linear network of oscillators, a nonlinear system and a dynamical system governed by a Partial Differential Equation (PDE).