Adaptive Data-Driven Probabilistic Reduced-Order Models for Parameterized Dynamical Systems

In this work, we propose a data-driven technique for constructing reduced-order models (ROMs) for parameterized nonlinear dynamical systems under a probabilistic framework. We use Gaussian process regression (GPR) to learn the nonlinear flow map of the dynamical system in the reduced-order space, the latter being identified with proper orthogonal decomposition (POD). This results in a dynamic mode decomposition (DMD)-like ROM that allows for a probabilistic prediction of the future trajectory of the dynamic model beyond the sampled domain. At an unsampled trial parameter, the low-dimensional states are estimated by interpolating the existing ones at the sampled parameters probabilistically, thereby allowing for parameterization of the ROM with natural uncertainty estimates in time, in space, and in the parameter domain. An adaptive sampling strategy is further proposed to optimize the selection of the parameter locations, where the snapshots are collected. The effectiveness of the proposed method is demonstrated on a series of numerical examples, and the results show that the proposed method is effective for high-dimensional parameterized dynamical problems.