Reduced-space Gaussian Process Regression for Data-Driven Probabilistic Forecast of Chaotic Dynamical Systems

TL;DR

This paper introduces a data-driven reduced-order Gaussian Process Regression method for probabilistic forecasting of complex chaotic systems, effectively capturing uncertainty and adapting to high-dimensional dynamics.

Contribution

It develops a novel reduced-space GPR framework with an adaptive scheme for accurate probabilistic forecasts of high-dimensional chaotic systems.

Findings

Effective in forecasting Lorenz 96 and Kuramoto-Sivashinsky systems

Quantifies local uncertainty in predictions

Performs well in highly turbulent regimes

Abstract

We formulate a reduced-order strategy for efficiently forecasting complex high-dimensional dynamical systems entirely based on data streams. The first step of our method involves reconstructing the dynamics in a reduced-order subspace of choice using Gaussian Process Regression (GPR). GPR simultaneously allows for reconstruction of the vector field and more importantly, estimation of local uncertainty. The latter is due to i) local interpolation error and ii) truncation of the high-dimensional phase space. This uncertainty component can be analytically quantified in terms of the GPR hyperparameters. In the second step we formulate stochastic models that explicitly take into account the reconstructed dynamics and their uncertainty. For regions of the attractor which are not sufficiently sampled for our GPR framework to be effective, an adaptive blended scheme is formulated to enforce correct statistical steady state properties, matching those of the real data. We examine the effectiveness of the proposed method to complex systems including the Lorenz 96, the Kuramoto-Sivashinsky, as well as a prototype climate model. We also study the performance of the proposed approach as the intrinsic dimensionality of the system attractor increases in highly turbulent regimes.