Koopman operators are infinite‐dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite‐dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data‐driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we compute smoothed approximations of spectral measures associated with general measure‐preserving dynamical systems. We prove explicit convergence theorems for our algorithms (including for general systems that are not measure‐preserving), which can achieve high‐order convergence even for chaotic systems when computing the density of the continuous spectrum and the discrete spectrum. Since our algorithms have error control, ResDMD allows aposteri verification of spectral quantities, Koopman mode decompositions, and learned dictionaries. We demonstrate our algorithms on the tent map, circle rotations, Gauss iterated map, nonlinear pendulum, double pendulum, and Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high‐dimensional state space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule with a 20,046‐dimensional state space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number >105that has a 295,122‐dimensional state space.
This content will become publicly available on May 31, 2024
Risk of Stochastic Systems for Temporal Logic Specifications
The wide availability of data coupled with the computational advances in artificial intelligence and machine learning promise to enable many future technologies such as autonomous driving. While there has been a variety of successful demonstrations of these technologies, critical system failures have repeatedly been reported. Even if rare, such system failures pose a serious barrier to adoption without a rigorous risk assessment. This article presents a framework for the systematic and rigorous risk verification of systems. We consider a wide range of system specifications formulated in signal temporal logic (STL) and model the system as a stochastic process, permitting discrete-time and continuous-time stochastic processes. We then define the STL robustness risk as the risk of lacking robustness against failure . This definition is motivated as system failures are often caused by missing robustness to modeling errors, system disturbances, and distribution shifts in the underlying data generating process. Within the definition, we permit general classes of risk measures and focus on tail risk measures such as the value-at-risk and the conditional value-at-risk. While the STL robustness risk is in general hard to compute, we propose the approximate STL robustness risk as a more tractable notion that upper bounds the STL robustness risk. We show how the approximate STL robustness risk can accurately be estimated from system trajectory data. For discrete-time stochastic processes, we show under which conditions the approximate STL robustness risk can even be computed exactly. We illustrate our verification algorithm in the autonomous driving simulator CARLA and show how a least risky controller can be selected among four neural network lane-keeping controllers for five meaningful system specifications.
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- Award ID(s):
- 2038873
- NSF-PAR ID:
- 10441191
- Date Published:
- Journal Name:
- ACM Transactions on Embedded Computing Systems
- Volume:
- 22
- Issue:
- 3
- ISSN:
- 1539-9087
- Page Range / eLocation ID:
- 1 to 31
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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