Physics-Guided Machine Learning Variational Multiscale Reduced Order Models*

The variational multiscale framework (VMS) is motivated by the locality of energy transfer and enabled by the hierarchy of the underlying structures. Thus, VMS appears to be a natural solution for the closure problem in projection-based reduced order models (ROMs). Applications of VMS in ROM often focus on the use of phenomenological closure models by analogy with finite elements and large eddy simulations. Recently, data-driven VMS-ROMs have been explored, where a polynomial-like structure was considered to represent the interactions between different scales. We extend this by investigating whether neural networks can reveal the nonlinear processes and correlations as represented by the mutual interactions between various batches of resolved and unresolved scales. Moreover, we embed the locality of energy transfer into the learning and inference process of the neural network in a physics-guided machine learning (PGML) framework. We showcase the applicability of the proposed PGML-VMS-ROM using a set of prototypical flow problems with strong nonlinearity.