Probabilistic forecast of nonlinear dynamical systems with uncertainty quantification*

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 a few numerical examples, including the time-dependent adiabatic GW (TD-aGW) method for understanding quantum many-body systems far from equilibrium, and Lorenz 96 model for simulating chaotic behaviors in nonlinear dynamical systems. The examples indicate that uncertainty of forecast can be properly quantified, whereas model or input misspecification can degrade the accuracy of uncertainty quantification.