Nonparametric forecasting of low-dimensional dynamical systems

TL;DR

This paper introduces a non-parametric method for forecasting low-dimensional stochastic dynamical systems using diffusion maps, enabling uncertainty quantification and application to complex systems like Lorenz and climate data.

Contribution

The paper develops a diffusion map-based non-parametric forecasting approach that converges to a Galerkin projection, allowing uncertainty quantification in equation-free dynamical modeling.

Findings

Successfully applied to stochastic differential equations on a torus

Accurately forecasted Lorenz system dynamics

Effectively modeled Niño-3.4 climate data

Abstract

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 Ni\~{n}o-3.4 data set which is used as a proxy of the El-Ni\~{n}o Southern Oscillation.