# Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization

arXiv: https://arxiv.org/abs/2303.01483 · HTML: https://mlsanity.com/paper/2303.01483/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 92 references that are in the map.
- First posted on arXiv 2 Mar 2023.

## Builds on (most cited first, 30 of 32)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Active Learning of Dynamics for Data-Driven Control Using Koopman Operators](https://mlsanity.com/paper/1906.05194/) (2019)
- [Koopman operator-based model reduction for switched-system control of PDEs](https://mlsanity.com/paper/1710.06759/) (2017)
- [Learning Dynamical Systems with Side Information](https://mlsanity.com/paper/2008.10135/) (2020)
- [Koopman-Based Neural Lyapunov Functions for General Attractors](https://mlsanity.com/paper/2203.12303/) (2022)
- [Convex computation of the maximum controlled invariant set for polynomial control systems](https://mlsanity.com/paper/1303.6469/) (2013)
- [Region of Attraction Estimation Using Invariant Sets and Rational Lyapunov Functions](https://mlsanity.com/paper/1604.01084/) (2016)
- [Computing controlled invariant sets from data using convex optimization](https://mlsanity.com/paper/1912.03256/) (2019)
- [Converging outer approximations to global attractors using semidefinite programming](https://mlsanity.com/paper/2005.03346/) (2020)
- [Estimation of consistent parameter sets for continuous-time nonlinear systems using occupation measures and LMI relaxations](https://mlsanity.com/paper/1303.4615/) (2013)
- [Probabilistic and Set-based Model Invalidation and Estimation using LMIs](https://mlsanity.com/paper/1311.7099/) (2013)
- [Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization](https://mlsanity.com/paper/1512.05599/) (2015)
- [Bounding stationary averages of polynomial diffusions via semidefinite programming](https://mlsanity.com/paper/1606.01153/) (2016)
- [Bounding averages rigorously using semidefinite programming: mean moments of the Lorenz system](https://mlsanity.com/paper/1610.05335/) (2016)
- [Rigorous bounds on the stationary distributions of the chemical master equation via mathematical programming](https://mlsanity.com/paper/1702.05468/) (2017)
- [Semidefinite Approximations of Reachable Sets for Discrete-time Polynomial Systems](https://mlsanity.com/paper/1703.05085/) (2017)
- [Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems](https://mlsanity.com/paper/1705.07096/) (2017)
- [Semidefinite Approximations of Invariant Measures for Polynomial Systems](https://mlsanity.com/paper/1807.00754/) (2018)
- [Convex computation of extremal invariant measures of nonlinear dynamical systems and Markov processes](https://mlsanity.com/paper/1807.08956/) (2018)
- [Bounding extreme events in nonlinear dynamics using convex optimization](https://mlsanity.com/paper/1907.10997/) (2019)
- [Minimum wave speeds in monostable reaction-diffusion equations: sharp bounds by polynomial optimization](https://mlsanity.com/paper/2006.05457/) (2020)
- [Peak Estimation and Recovery with Occupation Measures](https://mlsanity.com/paper/2009.06120/) (2020)
- [Peak Estimation for Uncertain and Switched Systems](https://mlsanity.com/paper/2103.13017/) (2021)
- [Data-Driven Optimal Control via Linear Transfer Operators: A Convex Approach](https://mlsanity.com/paper/2202.01856/) (2022)

## Built on by (most cited first, 2 of 2)

- [Learning graphons from data: Random walks, transfer operators, and spectral clustering](https://mlsanity.com/paper/2507.18147/) (2025)
- [Weighted Birkhoff Averages Accelerate Data-Driven Methods](https://mlsanity.com/paper/2511.17772/) (2025)

## Abstract (arXiv)

We present a flexible data-driven method for dynamical system analysis that does not require explicit model discovery. The method is rooted in well-established techniques for approximating the Koopman operator from data and is implemented as a semidefinite program that can be solved numerically. Furthermore, the method is agnostic of whether data is generated through a deterministic or stochastic process, so its implementation requires no prior adjustments by the user to accommodate these different scenarios. Rigorous convergence results justify the applicability of the method, while also extending and uniting similar results from across the literature. Examples on discovering Lyapunov functions, performing ergodic optimization, and bounding extrema over attractors for both deterministic and stochastic dynamics exemplify these convergence results and demonstrate the performance of the method.