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

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.

Submitted to Physica D: Nonlinear Phenomena

References in corpus (10)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Nonparametric forecasting of low-dimensional dynamical systems
• Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model
• Reduced models in chemical kinetics via nonlinear data-mining
• Least Squares Shadowing Sensitivity Analysis of a Modified Kuramoto-Sivashinsky Equation
• A probabilistic decomposition-synthesis method for the quantification of rare events due to internal instabilities
• Probabilistic description of extreme events in intermittently unstable systems excited by correlated stochastic processes
• Probabilistic response and rare events in Mathieu's equation under correlated parametric excitation
• Forecasting Turbulent Modes with Nonparametric Diffusion Models: Learning from noisy data
• Manifold learning techniques and model reduction applied to dissipative PDEs

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• Time Series Forecasting Using Manifold Learning
• Data-driven nonintrusive reduced order modeling for dynamical systems with moving boundaries using Gaussian process regression
• Conditional Gaussian Nonlinear System: a Fast Preconditioner and a Cheap Surrogate Model For Complex Nonlinear Systems
• Sequential Bayesian experimental design for estimation of extreme-event probability in stochastic dynamical systems
• A physics-informed neural network method for the approximation of slow invariant manifolds for the general class of stiff systems of ODEs
• Learning the Latent dynamics of Fluid flows from High-Fidelity Numerical Simulations using Parsimonious Diffusion Maps