Machine Learning-Based Quadratic Closures for Non-Intrusive Reduced Order Models

In the present work, we introduce a data-driven approach to enhance the accuracy of non-intrusive reduced order models (ROMs). In particular, we focus on ROMs built using proper orthogonal decomposition (POD) in an under-resolved and marginally-resolved regime, i.e., when the number of modes employed is not enough to capture the system dynamics. We propose a method to re-introduce the contribution of neglected modes through a quadratic correction term, given by the action of a quadratic operator on the POD coefficients. Differently from the state-of-the-art methodologies, where the operator is learned via least-squares optimization [R. Geelen, S. Wright, and K. Willcox, Comput. Methods Appl. Mech. Engrg., 403 (2023), 115717; J. Barnett and C. Farhat, J. Comput. Phys., 464 (2022), 111348], we propose to parametrize the operator by a Multi-Input Operators Network (MIONet). This way, we are able to build models with higher generalization capabilities, where the operator itself is continuous in space—thus agnostic of the domain discretization—and parameter-dependent. We test our model on two standard benchmarks in fluid dynamics and show that the correction term improves the accuracy of standard POD-based ROMs.