Complex systems with dynamics evolving on multiple timescales pose a tremendous challenge for data analysis and modeling. In practice, complex systems such as oceans and climate, when they exhibit clear scale separation, are often modeled by treating fast variables as stochastic or noise effects. Kernel algorithms in machine learning are powerful modeling tools that provide data-driven approximations of operators governing the macroscopic observables for these systems. We apply diffusion kernels to a deterministic slow-fast system that resembles a lower-dimensional SDE in the slow variable. The resulting diffusion eigenvectors computed from the deterministic slow variable are shown to approximate the analytic Laplace-Bertrami eigenfunctions of the lower-dimensional SDE. Using these eigenfunctions as a basis, we construct an operator semigroup modeling the slow dynamics, and study its predictive skill.
Back to Long Programs