Credible real-time simulation is a crucial enabler for digital twin technology, and data-driven model reduction is a key approach to achieving this. In this talk, we will discuss non-intrusive, probabilistic methods for learning reduced-order representations of high-dimensional dynamical systems, with built-in quantification of modeling uncertainties to certify computational reliability. The core strategy involves using Bayesian inference for the parametrization inspired by projection-based model reduction. Recently, we developed a novel method that leverages Gaussian processes approximations to formulate differential-equation-constrained likelihood functions and hence improve predictive performance, particularly when training data are noisy and/or scarce. This technique has demonstrated its effectiveness in data-driven reduced-order modeling by delivering accurate temporal predictions along with robust uncertainty quantification. This involves joint work with S. A. McQuarrie (Sandia National Laboratories), K. E. Willcox (UT Austin), and A. Chaudhuri (UT Austin), as well as with D. Ye (UTwente).