# Finite-data error bounds for Koopman-based prediction and control

arXiv: https://arxiv.org/abs/2108.07102 · HTML: https://mlsanity.com/paper/2108.07102/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Aug 2021.

## Built on by (most cited first, 26 of 26)

- [Learning invariant representations of time-homogeneous stochastic dynamical systems](https://mlsanity.com/paper/2307.09912/) (2023)
- [Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems](https://mlsanity.com/paper/2308.10697/) (2023)
- [Koopman-informed recurrent neural networks](https://mlsanity.com/paper/2410.23467/) (2024)
- [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024)
- [Self-Supervised Evolution Operator Learning for High-Dimensional Dynamical Systems](https://mlsanity.com/paper/2505.18671/) (2025)
- [Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories](https://mlsanity.com/paper/2209.09977/) (2022)
- [Learning a model is paramount for sample efficiency in reinforcement learning control of PDEs](https://mlsanity.com/paper/2302.07160/) (2023)
- [Equivariance and partial observations in Koopman operator theory for partial differential equations](https://mlsanity.com/paper/2307.15325/) (2023)
- [SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systems](https://mlsanity.com/paper/2402.03145/) (2024)
- [Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds](https://mlsanity.com/paper/2603.15091/) (2026)
- [Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr"om method](https://mlsanity.com/paper/2403.02811/) (2024)
- [Data-Driven Stochastic Optimal Control in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2407.16407/) (2024)
- [Data-driven Modeling in Metrology -- A Short Introduction, Current Developments and Future Perspectives](https://mlsanity.com/paper/2406.16659/) (2024)
- [End-to-end guarantees for indirect data-driven control of bilinear systems with finite stochastic data](https://mlsanity.com/paper/2409.18010/) (2024)
- [Kernel-Based Optimal Control: An Infinitesimal Generator Approach](https://mlsanity.com/paper/2412.01591/) (2024)
- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [Featurizing Koopman Mode Decomposition For Robust Forecasting](https://mlsanity.com/paper/2312.09146/) (2023)
- [Data-Driven Predictive Control of Nonholonomic Robots Based on a Bilinear Koopman Realization: Data Does Not Replace Geometry](https://mlsanity.com/paper/2411.07192/) (2024)
- [Automatic feature identification in least-squares policy iteration using the Koopman operator framework](https://mlsanity.com/paper/2603.26464/) (2026)
- [Hamiltonian Graph Inference Networks: Joint structure discovery and dynamics prediction for lattice Hamiltonian systems from trajectory data](https://mlsanity.com/paper/2604.23606/) (2026)
- [Dictionary learning for Kernel EDMD](https://mlsanity.com/paper/2604.25572/) (2026)
- [Finding Koopman Invariant Subspaces via Personalized PageRank](https://mlsanity.com/paper/2605.24666/) (2026)
- [Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training error](https://mlsanity.com/paper/2605.31438/) (2026)
- [Reward-free Pretraining for Reinforcement Learning via Occupancy Coverage Maximization](https://mlsanity.com/paper/2606.21271/) (2026)
- [Learning to control switching nonlinear systems with Koopman operator regression](https://mlsanity.com/paper/2607.11344/) (2026)
- [Demystifying Linear Operator Learning for Control Systems](https://mlsanity.com/paper/2609.19428/) (2026)

## Abstract (arXiv)

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.