  Modeling geophysical processes as low-dimensional dynamical systems and
regressing their vector field from data is a promising approach for learning
emulators of such systems. We show that when the kernel of these emulators is
also learned from data (using kernel flows, a variant of cross-validation),
then the resulting data-driven models are not only faster than equation-based
models but are easier to train than neural networks such as the long short-term
memory neural network. In addition, they are also more accurate and predictive
than the latter. When trained on geophysical observational data, for example,
the weekly averaged global sea-surface temperature, considerable gains are also
observed by the proposed technique in comparison to classical partial
differential equation-based models in terms of forecast computational cost and
accuracy. When trained on publicly available re-analysis data for the daily
temperature of the North-American continent, we see significant improvements
over classical baselines such as climatology and persistence-based forecast
techniques. Although our experiments concern specific examples, the proposed
approach is general, and our results support the viability of kernel methods
(with learned kernels) for interpretable and computationally efficient
geophysical forecasting for a large diversity of processes.
