# Koopman operator-based model reduction for switched-system control of PDEs

arXiv: https://arxiv.org/abs/1710.06759 · HTML: https://mlsanity.com/paper/1710.06759/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Oct 2017.

## Built on by (most cited first, 30 of 31)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Distributed Control of Partial Differential Equations Using Convolutional Reinforcement Learning](https://mlsanity.com/paper/2301.10737/) (2023)
- [PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator](https://mlsanity.com/paper/2306.12962/) (2023)
- [Adversarial dynamical systems characterize when data-driven learning succeeds or fails](https://mlsanity.com/paper/2407.06312/) (2024)
- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024)
- [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)
- [On the Universal Transformation of Data-Driven Models to Control Systems](https://mlsanity.com/paper/2102.04722/) (2021)
- [Generalizing Dynamic Mode Decomposition: Balancing Accuracy and Expressiveness in Koopman Approximations](https://mlsanity.com/paper/2108.03712/) (2021)
- [Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order models](https://mlsanity.com/paper/2409.05709/) (2024)
- [Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition](https://mlsanity.com/paper/2207.07719/) (2022)
- [Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization](https://mlsanity.com/paper/2303.01483/) (2023)
- [Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr"om method](https://mlsanity.com/paper/2403.02811/) (2024)
- [Solving Partial Differential Equations with Equivariant Extreme Learning Machines](https://mlsanity.com/paper/2404.18530/) (2024)
- [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)
- [Policy Optimization for PDE Control with a Warm Start](https://mlsanity.com/paper/2403.01005/) (2024)
- [Data-driven system identification using quadratic embeddings of nonlinear dynamics](https://mlsanity.com/paper/2501.08202/) (2025)
- [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (2026)
- [Deep Learning for Stability Analysis of a Freely Vibrating Sphere at Moderate Reynolds Number](https://mlsanity.com/paper/2112.09858/) (2021)
- [Semi-Supervised Learning of Dynamical Systems with Neural Ordinary Differential Equations: A Teacher-Student Model Approach](https://mlsanity.com/paper/2310.13110/) (2023)
- [Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices](https://mlsanity.com/paper/2410.02199/) (2024)
- [Latent feedback control of distributed systems in multiple scenarios through deep learning-based reduced order models](https://mlsanity.com/paper/2412.09942/) (2024)
- [Nonlinear Model Order Reduction of Dynamical Systems in Process Engineering: Review and Comparison](https://mlsanity.com/paper/2506.12819/) (2025)
- [Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2509.22482/) (2025)
- [Learning Biomolecular Motion: The Physics-Informed Machine Learning Paradigm](https://mlsanity.com/paper/2511.06585/) (2025)
- [Learning to control switching nonlinear systems with Koopman operator regression](https://mlsanity.com/paper/2607.11344/) (2026)
- [Cluster-Weighted EDMD](https://mlsanity.com/paper/2607.12243/) (2026)
- [Real-time optimal control with shallow recurrent decoder networks](https://mlsanity.com/paper/2607.19302/) (2026)

## Abstract (arXiv)

We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. In situations where the Koopman operator can be computed exactly using Extended Dynamic Mode Decomposition (EDMD), the proposed approach yields optimal control inputs. Furthermore, a recent convergence result for EDMD suggests that the approach can be applied to more complex dynamics as well. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier--Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.