Probabilistic forecast of nonlinear dynamical systems with uncertainty quantification

TL;DR

This paper develops a probabilistic framework for forecasting nonlinear dynamical systems, integrating Gaussian processes and dynamic mode decomposition to quantify uncertainty and improve predictive reliability.

Contribution

It introduces an extension of Gaussian processes for vector-valued predictions and links dynamic mode decomposition with probabilistic models, providing a unified approach for uncertainty quantification in nonlinear dynamics.

Findings

Uncertainty can be effectively quantified in forecasts.

Model misspecification degrades uncertainty estimates.

The approach applies to systems with known and unknown inputs.

Abstract

Data-driven modeling is useful for reconstructing nonlinear dynamical systems when the underlying process is unknown or too expensive to compute. Having reliable uncertainty assessment of the forecast enables tools to be deployed to predict new scenarios unobserved before. In this work, we first extend parallel partial Gaussian processes for predicting the vector-valued transition function that links the observations between the current and next time points, and quantify the uncertainty of predictions by posterior sampling. Second, we show the equivalence between the dynamic mode decomposition and the maximum likelihood estimator of the linear mapping matrix in the linear state space model. The connection provides a {probabilistic generative} model of dynamic mode decomposition and thus, uncertainty of predictions can be obtained. Furthermore, we draw close connections between different data-driven models for approximating nonlinear dynamics, through a unified view of generative models. We study two numerical examples, where the inputs of the dynamics are assumed to be known in the first example and the inputs are unknown in the second example. The examples indicate that uncertainty of forecast can be properly quantified, whereas model or input misspecification can degrade the accuracy of uncertainty quantification.