Adaptive data-driven probabilistic reduced-order models for parameterized dynamical system

In this work, we propose a data-driven technique for constructing reduced-order models (ROMs) for parameterized nonlinear dynamical systems under a probabilistic framework. Specifically, we advocate the use of Gaussian process regression (GPR) with various kernels to learn the nonlinear flow map of the dynamical system under consideration; the underlying reduced-order subspace is identified by the method of proper orthogonal decomposition (POD). This results in a dynamic mode decomposition (DMD)-like ROM that allows for a probabilistic forecast of the unsampled, future trajectory of the dynamic model. At an unsampled trial parameter, the ROM is constructed by interpolating a set of sampled ROMs probabilistically, thereby allowing for parameterization of the ROM with intrinsic 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 propose method is demonstrated on a series of numerical examples of increasing sophistication.