A reduced-order model based on Gaussian process dynamical models for time-dependent parameterized partial differential equations

A reduced-order modeling framework is developed to address the high-dimensional challenges of parameterized partial differential equations by integrating tensor-train decomposition (TTD), Gaussian process regression (GPR), and Gaussian process dynamical models (GPDMs). TTD furnishes a low-rank approximation of the solution snapshots, while GPR learns the nonlinear mapping from the input parameter space to the tensor-train format. GPDM then models the temporal dynamics, enabling accurate time evolution prediction and uncertainty quantification. The method is validated on several benchmark problems, including Burgers' equations and the incompressible Navier-Stokes equations. Comparative experiments against traditional methods such as proper orthogonal decomposition-Gaussian process regression and dynamic mode decomposition based on tensor-train decomposition-Gaussian process regression demonstrate that the proposed approach achieves superior accuracy in modeling nonlinear temporal dynamics, conducting time-domain interpolation, and quantifying prediction uncertainty.