Over the past three decades, operator-theoretic techniques have proven to be highly fruitful for analysis and data-driven modeling of dynamical systems. These methods leverage the linearity of the action of (nonlinear) dynamics on spaces of observables or probability distributions (induced by Koopman or transfer operators, respectively) to perform spectral decomposition, forecasting, and uncertainty quantification in complex systems. In this talk, we survey aspects of the mathematical formulation of data-driven dynamical operator methods and present applications to climate dynamics on daily to interannual timescales. Central to our approach are kernel methods for consistent operator approximation in both supervised and unsupervised learning problems. We illustrate these methods with applications to extraction and prediction of the El Nino Southern Oscillation and dynamical closure in cloud-resolving atmospheric models.
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