On Computation of Koopman Operator from Sparse Data

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.

Optimal reporter placement in sparsely measured genetic networks using the Koopman operator

“…In the presence of sparse and noisy data, [21] showed that the Koopman learning problem can be formulated as a robust optimization problem, which is equivalent to a specific regularized learning problem in which the LASSO penalty parameter corresponds to the upper bound on the noise. We will suppose, for simplicity of exposition of the technique, that an exact Koopman operator for the coarse-time step mapping t to t+N is either known or obtained directly from data satisfying…”

“…This provides a powerful scheme for estimating the governing equations of a fine-grained time-evolving biological process from sparse or coarse-grained temporal measurements, so long as the coarse-grained time measurement is a product of regularly spaced intervals of time in the fine-grained representation. Again, since RNAseq and proteomic measurements often provide full-state measurements of a network, this in theory can provide sufficient information to recover the Koopman operator, even in the presence of noise [21]. The key insight and property leveraged is the linearity of the lifted Koopman representation.…”

Data-Driven Operator Theoretic Methods for Global Phase Space Learning

“…The Koopman operator definition is an infinite-dimensional linear operator that yields the evolution of a dynamical system on an infinite-dimensional space of functions. Finding an infinite dimensional (linear) operator is computationally infeasible, hence many approaches are devised to best approximate the Koopman operator in finite dimensional space [3,33,34,12,35,36,37,38,39,40,41]. Most popular methods include dynamic mode decomposition (DMD) [3,33]; extended dynamic mode decomposition (E-DMD) [12,42]; kernel dynamic mode decomposition (K-DMD) [34], naturally structured dynamic mode decomposition (NS-DMD) [43], Hankel-DMD [17], deep dynamic mode decomposition (deep-DMD) [6,42,44,45,46].…”

“…Koopman operator is an infinite dimensional operator that can represent the evolution of a dynamical system. However, finding an infinite dimensional (linear) operator is computationally intractable and numerous methods have been developed to best represent the (nonlinear) system dynamics with a finite dimensional Koopman operator [3], [14]- [19]. Most popular methods include, dynamic mode decomposition (DMD) [3], [14]; extended dynamic mode decomposition (E-DMD) [16], [20]; kernel dynamic mode decomposition (K-DMD) [15], naturally structured dynamic mode decomposition (NS-DMD) [21], deep dynamic mode decomposition (deep-DMD) [6], [22].…”

Towards Scalable Koopman Operator Learning: Convergence Rates and A Distributed Learning Algorithm

“…The power of these operator theoretic methods is that it provides linear representations of nonlinear time-invariant systems, albeit in higher dimensional spaces that are sometimes countable or uncountable. Various numerical approaches, such as dynamic mode decomposition(DMD), Hankel-DMD, extended dynamic mode decomposition (E-DMD), structured dynamic mode decomposition (S-DMD) have been proposed for discovering the Koopman operator of a nonlinear system, using a series of dictionary functions with spanning or universal function approximation properties [2]- [6]. Recently, researchers have shown it is possible to integrate machine-driven learning representations with dynamic mode decomposition algorithms, using variational autoencoders to achieve phase-dependent representations of spectra [7] or delay embeddings [8], shallow neural networks [3], linearly recurrent neural networks for balancing expressiveness and overfitting [9], and deep RELU feedforward networks for predictive modeling in biological and transmission systems [10].…”