Title: Reduced Modeling of Unknown Trajectories,
Abstract: This paper deals with model order reduction of parametrical dynamical
systems. We consider the specific setup where the distribution of the system's
trajectories is unknown but the following two sources of information are
available: \textit{(i)} some "rough" prior knowledge on the system's
realisations; \textit{(ii)} a set of "incomplete" observations of the system's
trajectories. We propose a Bayesian methodological framework to build
reduced-order models (ROMs) by exploiting these two sources of information. We
emphasise that complementing the prior knowledge with the collected data
provably enhances the knowledge of the distribution of the system's
trajectories. We then propose an implementation of the proposed methodology
based on Monte-Carlo methods. In this context, we show that standard ROM
learning techniques, such e.g. Proper Orthogonal Decomposition or Dynamic Mode
Decomposition, can be revisited and recast within the probabilistic framework
considered in this paper.~We illustrate the performance of the proposed
approach by numerical results obtained for a standard geophysical model.