On Computation of Koopman Operator from Sparse Data

TL;DR

This paper introduces a robust optimization-based method to compute the Koopman operator accurately from sparse data by augmenting data with artificial points and handling noise, demonstrated on various dynamical systems.

Contribution

It presents a novel algorithm that enables Koopman operator approximation from limited data using artificial data augmentation and robust optimization techniques.

Findings

Effective in linear, nonlinear, and PDE systems

Outperforms traditional methods with sparse data

Robust to noise and data limitations

Abstract

In this paper we propose a novel approach to compute the Koopman operator from sparse time series data. In recent years there has been considerable interests in operator theoretic methods for data-driven analysis of dynamical systems. Existing techniques for the approximation of the Koopman operator require sufficiently large data sets, but in many applications, the data set may not be large enough to approximate the operators to acceptable limits. In this paper, using ideas from robust optimization, we propose an algorithm to compute the Koopman operator from sparse data. We enrich the sparse data set with artificial data points, generated by adding bounded artificial noise and and formulate the noisy robust learning problem as a robust optimization problem and show that the optimal solution is the Koopman operator with smallest error. We illustrate the efficiency of our proposed approach in three different dynamical systems, namely, a linear system, a nonlinear system and a dynamical system governed by a partial differential equation.