Reduced-space Gaussian Process Regression for data-driven probabilistic forecast of chaotic dynamical systems

Abstract

We formulate a reduced-order data-driven strategy for the efficient probabilistic forecast of complex high-dimensional dynamical systems for which data-streams are available. The first step of our method consists of the reconstruction of the vector field in a reduced-order subspace of interest using Gaussian Process Regression (GPR). GPR simultaneously allows for the reconstruction of the vector field, as well as the estimation of the local uncertainty. The latter is due to i) the local interpolation error and ii) due to the truncation of the highdimensional phase space and it analytically quantified in terms of the GPR hyperparameters. The second step involves the formulation of stochastic models that explicitly take into account the reconstructed dynamics and their uncertainty. For regions of the attractor where the training data points are not sufficiently dense for GPR to be effective an adaptive blended scheme is formulated that guarantess correct statistical steady state properties. 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.

Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks

“…This may be attributed to the absence of certain attractor regions in the training data, insufficient training, and propagation of the exponentially increasing prediction error. To mitigate this effect, LSTM is also combined with MSM, following ideas presented in [36], in order to guarantee convergence to the invariant measure. Blending LSTM or GPR with MSM leads to a deterioration in the short-term prediction performance but the steady-state statistical behavior is captured.…”

“…In the following we refer to this model as the Lorenz 96. The Lorenz 96 is usually used ( [57,36] and references therein) as a toy problem to benchmark methods for weather prediction. The system of differential equations that governs the Lorenz 96 is defined as…”

Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems

“…During the evolution, the energy input and dissipation assume smaller values and are very close to each other sitting near the diagonal. The Kolmogorov flow is driven by the external forcing f such that growth in the energy input I corresponds to the alignment of the velocity field u and the forcing (see equation (15)). This alignment leads to an abrupt increase in the energy input I. Consequently, the energy dissipation also increases bringing the trajectory back to the statistically stationary background.…”

“…However, for high-dimensional chaotic attractors the reconstructed dynamics have a poor forecasting skill (see e.g. [15,16]) which is comparable with Mean Square Models (models based on carefully tuned Langevin equations [17]). Since rare extreme events are associated with strong nonlinearities and intermittently positive Lyapunov exponents (i.e., high sensitivity to perturbations), their prediction from a finite set of observations is challenging and remains an active area of research (see, e.g., Giannakis and Majda [18], Bialonski et al [19]).…”

Forecast and evaluation of COVID-19 spreading in USA with reduced-space Gaussian process regression

“…• With Bayesian inference and Markov chain Monte Carlo algorithm, they have demonstrated more accuracy in forecast associated to COVID-19 [10] . • Associated to GPR the idea is to "utilize nonlinear diffusion map coordinates and formulate a deterministic dynamical system on the system manifold" [12] . A interesting approach has a reduced-space data-driven dynamical system with an evaluation, with efficiency with low intrinsic dimensionality.…”