On Computation of Koopman Operator from Sparse Data

In this paper, we propose a novel approach to compute the Koopman operator from sparse time series data. In recent years there have 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 formulate the noisy robust learning problem as a robust optimization problem and show that the optimal solution is the Koopman operator with the 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.