Assessing how complex systems respond to external perturbations is a fundamental problem in physics, climatology, and engineering. Direct perturbation is often impossible: we have only one realization of Earth's climate or a single time series of observations. Classical linear response theory requires knowledge of the dynamical equations, which is rarely available for real data. This work sits among methods for extracting Koopman operators from data (e.g., neighboring works on forecasting and noise analysis), but focuses on optimizing the response rather than on prediction or identification. This fills a gap between perturbation theory and the practice of analyzing high-dimensional observations.
The authors use kernel smoothing to approximate the Koopman operator from an observed trajectory. When necessary, Takens' delay embedding is used to reconstruct hidden states. The resulting transition matrix of a finite Markov chain approximates the dynamics. Then an optimization problem is solved: find a perturbation matrix with zero row sums and constraints on allowed transitions that maximizes the derivative of the modulus or argument of a chosen eigenvalue. The solution is given by an explicit formula involving the left and right eigenvectors. For visualization, an 'optimal response vector field' is introduced, constructed by multiplying the perturbation matrix by the observation matrix.
For the example of nonuniform rotation on a circle, the optimal perturbation to speed up the cycle turned out to be tangential to the trajectory and inversely proportional to the current speed. For the Lorenz system, the perturbation that accelerates rotation around the wings is orthogonal to the level lines of the argument of the complex eigenvector, not along the flow direction. In the El Niño climate example, the optimal perturbation to increase frequency leads the cycle by π/2, while that to strengthen correlations is in phase. A connection with the Indian Ocean Dipole phase was found: negative IOD during El Niño buildup and positive during La Niña.
This work differs from neighboring methods in that it is the first to allow optimizing the response in observation space without knowing the dynamical equations. Neighboring approaches either predict the future or extract patterns, but do not seek a targeted perturbation. Here, optimization is performed over the Koopman operator spectrum, yielding physically interpretable vector fields. This expands the toolkit for analyzing complex systems, especially in climatology, where experiments are impossible.
The method can be extended to general linear response problems, not only spectral properties. It may also apply to stochastic systems and nonstationary data. Investigating the robustness of optimal perturbations to the choice of kernel parameters and delay is promising.
The method could influence climate modeling, El Niño forecasting, control of chaotic systems, and analysis of biological rhythms.
The next step will be to test the method on real climate data and compare with direct model experiments.
This work is directly related to the problem of identifying and controlling complex systems from partial observations. In dynamical systems, the question of rigorous convergence conditions for kernel approximations on chaotic attractors remains open.
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What the work does
The authors construct a kernel-smoothed approximation of the Koopman operator from a time series of observations, possibly in a high-dimensional space, and formulate an optimization problem to find the infinitesimal perturbation that maximally changes selected spectral properties: the frequency or decay rate of almost-cycles, or the stability of almost-invariant sets. The method is illustrated with examples ranging from circle rotations to El Niño in a climate model. Explicit formulas for the optimal perturbation are derived, and an optimal response vector field is introduced.
Where its strength is
This work is the first to optimize the response in the observation space without knowing the dynamics equations, using the Koopman operator spectrum. The physical interpretability of the vector fields and the applicability to climate data make the approach valuable for analyzing complex systems where experiments are impossible.
Where it could go next
Worth trying
Why this may matter
If the proposed approach can be extended to more realistic noisy data and integrated with model selection methods, it could provide a practical tool for targeted intervention in complex systems, for example, to control climate modes or engineering systems without a full model.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.52, against a usual 0.64. It looks like few have drilled nearby.
To speed up rotation of a nonuniform circle, the optimal perturbation is stronger in slow regions: this follows from Jensen's inequality, which states that to minimize the time for one revolution, one should equalize the speed.