State Space Kriging model for emulating nonlinear stochastic dynamical systems with parameter uncertainty

We extend our recently proposed state space Kriging (S2K) model (Cheng et al., 2025), which approximates the system dynamics in state space with a sparse Kriging surrogate, to deal with system parameter uncertainty. We propose and investigate two methods. (1) A two-step method that employs the standard S2K model to approximate the conditional state space representation given a series of realizations of the uncertain system parameters. The surrogate model for arbitrary system parameter values is then approximated by interpolation. (2) A one-step method that represents the random system parameters by pseudo stochastic processes. This allows the direct application of the S2K framework to approximate the state space representation with uncertain system parameters. We numerically compare the performance of the two S2K methods with the state-of-the-art polynomial chaos nonlinear auto-regressive with exogenous input (PC-NARX) model for emulating the response of complex nonlinear stochastic dynamical systems. The results indicate that S2K results in more accurate predictions while reducing the required number of input training time histories by two orders of magnitude relative to PC-NARX. This work lays the foundation on uncertainty quantification application of large-scale state space surrogate model of nonlinear stochastic dynamical systems.