Abstract: For outer loop applications such as uncertainty quantification, control, and optimization, building certifiably accurate surrogate and reduced order models (ROM) is of particular importance, especially these outer loop goals are part of a digital twin infrastructure. We present recent work on two new approaches for surrogate model/ROM constructions that come with new and useful attractive versions of certifiability. The first approach is a ROM construction for parametric linear dynamical systems that provides both an a priori and a posteriori understanding of ROM error. The second approach addresses supervised operator learning that provides sample complexity estimates for stability and accuracy. Of particular interest is the ability of this approach to provide error in high-regularity norms, e.g., Sobolev norms. We will discuss promises and pitfalls of these approaches, and present an outlook toward adaptive constructions, uncertainty estimates, and nonlinear model reduction and surrogate construction.