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

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).