A quantitative analysis of Koopman operator methods for system identification and predictions

Abstract

We give convergence and cost estimates for a data-driven system identification method: given an unknown dynamical system, the aim is to recover its vector field and its flow from trajectory data. It is based on the so-called Koopman operator, which uses the well-known link between differential equations and linear transport equations. Data-driven methods recover specific finite-dimensional approximations of the Koopman operator, which can be understood as a transport operator. We focus on such approximations given by classical finite element spaces, which allow us to give estimates on the approximation of the Koopman operator as well as the solutions of the associated linear transport equation. These approximations are thus relevant objects to solve the system identification problem.We then analyze the convergence of a variant of the generator Extended Dynamic Mode Decomposition (gEDMD) algorithm, one of the main algorithms developed to compute approximations of the Koopman operator from data. We find however that, when combining this algorithm with classical finite element spaces, the results are not satisfactory numerically, as the convergence of the data-driven approximation is too slow for the method to benefit from the accuracy of finite element spaces. In particular, for problems in dimension 1 it is less efficient than direct interpolation methods to recover the vector field. We provide some numerical examples to illustrate this last point.

Finite-Data Error Bounds for Koopman-Based Prediction and Control

“…In case of an ODE, the set X is often assumed to be compact and forward-invariant and the probability measure is the standard Lebesgue measure, cf. Zhang and Zuazua (2021).…”

“…We briefly comment on this assumption and first note that forward invariance of X can be weakened, if one is only interested in estimates for states contained in X, see also Zhang and Zuazua (2021, Section 3.2). Moreover, if the dynamics obey an ODE, it was shown that the Koopman operator can indeed be extended to a strongly continuous semigroup on L 2 μ (X), see also Zhang and Zuazua (2021). Second, the assumption of invariance of the underlying probability measure is satisfied for a broad class of SDEs, see, e.g., Risken (1996).…”

“…8.5. In Zhang and Zuazua (2021), besides providing a finite-data error bound on the approximation of the Koopman operator in the context of ODEs, the authors estimate the projection error by means of finite-element analysis. In conclusion, to the best of our knowledge, 1 Zhang and Zuazua (2021), Kurdila and Bobade (2018) are the only works providing rigorous error bounds for Koopman-based approximations of a dynamical system governed by a nonlinear ODE.…”

“…The deduced bounds on the approximation error and prediction accuracy explicitly depend on the number of data points used in eDMD. To this end, besides using mass concentration inequalities and a numerical error analysis to deal with the error propagation in time, we employ substantially different techniques in comparison to Kurdila and Bobade (2018), Zhang and Zuazua (2021) to provide an additional alternative assumption based on ergodic sampling tailored to stationary SDEs. Our results in this setting focus on the concept of asymptotic variance, see Lelièvre and Stoltz (2016) and the references therein.…”

“…This serves as additional motivation to continue the study of the sampling error for ergodic sampling by means of asymptotic variances. Let us stress that we do not fully analyze the projection error; in other words, how much the Koopman generator fails to be invariant on the approximation subspace, referring to the existing literature Kurdila and Bobade (2018) and Zhang and Zuazua (2021) in the autonomous case and our follow-up work (Schaller et al 2022) in the control setting. W.r.t.…”