2014/11/30 by Tyrus Berry, Dimitrios Giannakis, John Harlim · 1 citation
Mathematics · #math.DS
paper · pdf · doi:10.1103/physreve.91.032915
published as Phys. Rev. E 91, 032915 (2015) · Supplemental videos available at: http://personal.psu.edu/thb11/
arxiv created 2015/01/14 · arxiv updated 2015/03/25
This letter presents a non-parametric modeling approach for forecasting stochastic dynamical systems on low-dimensional manifolds. The key idea is to represent the discrete shift maps on a smooth basis which can be obtained by the diffusion maps algorithm. In the limit of large data, this approach converges to a Galerkin projection of the semigroup solution to the underlying dynamics on a basis adapted to the invariant measure. This approach allows one to quantify uncertainties (in fact, evolve the probability distribution) for non-trivial dynamical systems with equation-free modeling. We verify our approach on various examples, ranging from an inhomogeneous anisotropic stochastic differential equation on a torus, the chaotic Lorenz three-dimensional model, and the Niño-3.4 data set which is used as a proxy of the El-Niño Southern Oscillation.