# Optimal construction of Koopman eigenfunctions for prediction and control

arXiv: https://arxiv.org/abs/1810.08733 · HTML: https://mlsanity.com/paper/1810.08733/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Oct 2018.

## Built on by (most cited first, 30 of 35)

- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Diffeomorphically Learning Stable Koopman Operators](https://mlsanity.com/paper/2112.04085/) (2021)
- [Koopman-Assisted Reinforcement Learning](https://mlsanity.com/paper/2403.02290/) (2024)
- [PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator](https://mlsanity.com/paper/2306.12962/) (2023)
- [Learning the Koopman Eigendecomposition: A Diffeomorphic Approach](https://mlsanity.com/paper/2110.07786/) (2021)
- [Task-Oriented Koopman-Based Control with Contrastive Encoder](https://mlsanity.com/paper/2309.16077/) (2023)
- [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025)
- [Koopman Spectrum Nonlinear Regulators and Efficient Online Learning](https://mlsanity.com/paper/2106.15775/) (2021)
- [Generalizing Dynamic Mode Decomposition: Balancing Accuracy and Expressiveness in Koopman Approximations](https://mlsanity.com/paper/2108.03712/) (2021)
- [SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systems](https://mlsanity.com/paper/2402.03145/) (2024)
- [Online Estimation of the Koopman Operator Using Fourier Features](https://mlsanity.com/paper/2212.01503/) (2022)
- [Information Shapes Koopman Representation](https://mlsanity.com/paper/2510.13025/) (2025)
- [Data-driven End-to-end Learning of Pole Placement Control for Nonlinear Dynamics via Koopman Invariant Subspaces](https://mlsanity.com/paper/2208.08883/) (2022)
- [Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition](https://mlsanity.com/paper/2207.07719/) (2022)
- [Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr"om method](https://mlsanity.com/paper/2403.02811/) (2024)
- [Koopman Eigenfunction-Based Identification and Optimal Nonlinear Control of Turbojet Engine](https://mlsanity.com/paper/2505.10438/) (2025)
- [A purely data-driven framework for prediction, optimization, and control of networked processes: application to networked SIS epidemic model](https://mlsanity.com/paper/2108.02005/) (2021)
- [Towards Data-driven LQR with Koopmanizing Flows](https://mlsanity.com/paper/2201.11640/) (2022)
- [Data-Driven Encoding: A New Numerical Method for Computation of the Koopman Operator](https://mlsanity.com/paper/2301.06542/) (2023)
- [Automated Global Analysis of Experimental Dynamics through Low-Dimensional Linear Embeddings](https://mlsanity.com/paper/2411.00989/) (2024)
- [Damage Adaptation in Seconds for Architected Materials](https://mlsanity.com/paper/2606.17394/) (2026)
- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [Learned Lifted Linearization Applied to Unstable Dynamic Systems Enabled by Koopman Direct Encoding](https://mlsanity.com/paper/2210.13602/) (2022)
- [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)
- [Federated Koopman-Reservoir Learning for Large-Scale Multivariate Time-Series Anomaly Detection](https://mlsanity.com/paper/2503.11255/) (2025)
- [Deep Neural Koopman Operator-based Economic Model Predictive Control of Shipboard Carbon Capture System](https://mlsanity.com/paper/2504.06818/) (2025)
- [Nonlinear Model Order Reduction of Dynamical Systems in Process Engineering: Review and Comparison](https://mlsanity.com/paper/2506.12819/) (2025)
- [KoopCast: Trajectory Forecasting via Koopman Operators](https://mlsanity.com/paper/2509.15513/) (2025)
- [Deep Koopman Economic Model Predictive Control of a Pasteurisation Unit](https://mlsanity.com/paper/2511.04437/) (2025)
- [Spatially Aware Dictionary-Free Koopman Eigenfunction Identification for Modeling and Control](https://mlsanity.com/paper/2511.22648/) (2025)

## Abstract (arXiv)

This work presents a novel data-driven framework for constructing eigenfunctions of the Koopman operator geared toward prediction and control. The method leverages the richness of the spectrum of the Koopman operator away from attractors to construct a rich set of eigenfunctions such that the state (or any other observable quantity of interest) is in the span of these eigenfunctions and hence predictable in a linear fashion. The eigenfunction construction is optimization-based with no dictionary selection required. Once a predictor for the uncontrolled part of the system is obtained in this way, the incorporation of control is done through a multi-step prediction error minimization, carried out by a simple linear least-squares regression. The predictor so obtained is in the form of a linear controlled dynamical system and can be readily applied within the Koopman model predictive control framework of [12] to control nonlinear dynamical systems using linear model predictive control tools. The method is entirely data-driven and based purely on convex optimization, with no reliance on neural networks or other non-convex machine learning tools. The novel eigenfunction construction method is also analyzed theoretically, proving rigorously that the family of eigenfunctions obtained is rich enough to span the space of all continuous functions. In addition, the method is extended to construct generalized eigenfunctions that also give rise Koopman invariant subspaces and hence can be used for linear prediction. Detailed numerical examples with code available online demonstrate the approach, both for prediction and feedback control.