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

Abstract

The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points, for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein–Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.

Learning a model is paramount for sample efficiency in reinforcement learning control of PDEs

“…At this point, suitable candidates appear to be GRUs instead of LSTMs (see, e.g., [54]), sparse regreesion techinques to identify governing equations [6], [55], and in particular the highly popular Koopman operator [56]- [58], as it allows us to learn linear models of nonlinear systems, which is very efficient both in terms of the required training data and the run time. Finally, it might be worth looking into recent prediction error results for these methods [59]- [61] and see whether they can be transferred into guarantees for the RL process.…”

Data-Driven Control of Nonlinear Systems: Beyond Polynomial Dynamics

“…In order to apply the proposed design scheme, we predefine the outer bound ∆ Φ on the safe operating region following the approach in [18, Procedure 7]. In particular, we first solve (15) for Qz = −I, Sz = 0, and arbitrary Rz > 0 without considering the second liner matrix inequality (16). This leads to a matrix P which relates the measured data and the chosen observables to infer a likely behavior of the closed-loop system, such that the safe operating region is maximized according to the underlying system dynamics.…”

“…Part I: x ∈ X SOR implies Φ(x) ∈ ∆ Φ : In order to represent the lifted dynamics via (40), we require Φ(x) ∈ ∆ Φ for all times. To this end, we exploit that (16) after dividing the inequality by ν. We rewrite this inequality as…”

“…To this end, [15] proved convergence of EDMD to the Koopman operator in the infinite-data limit. First finite-data error bounds, including bilinear EDMD for control systems, were proven in [16]. The error bounds in these work, however, are global in the sense that the error does not vanish at the origin, which prevents a straightforward application of standard control methods.…”