Machine Learning-based quadratic closures for non-intrusive Reduced Order Models

TL;DR

This paper introduces a novel data-driven quadratic correction method for non-intrusive reduced order models, using a neural network to improve accuracy and generalization in fluid dynamics simulations.

Contribution

It proposes a Multi-Input Operators Network to learn quadratic corrections, enhancing ROM accuracy and domain-agnostic capabilities compared to existing methods.

Findings

Quadratic correction improves ROM accuracy in fluid dynamics benchmarks.

The MIONet-based operator generalizes better across different domains.

The approach outperforms traditional least-squares based methods.

Abstract

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 optimisation, we propose to parametrise the operator by a Multi-Input Operators Network (MIONet). This way, we are able to build models with higher generalisation capabilities, where the operator itself is continuous in space -- thus agnostic of the domain discretisation -- 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.