# Reduced Modeling of Unknown Trajectories

## TL;DR

This paper introduces a Bayesian framework for reduced-order modeling of parametrical dynamical systems with unknown trajectory distributions, leveraging prior knowledge and incomplete observations to improve model accuracy.

## Contribution

It presents a novel probabilistic approach to model order reduction that incorporates prior knowledge and data, enhancing the understanding of system trajectories.

## Key findings

- Probabilistic ROMs outperform traditional methods in uncertain settings.
- Monte-Carlo methods effectively implement the Bayesian framework.
- Numerical results demonstrate improved modeling of geophysical systems.

## 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.