Neural network closures for nonlinear model order reduction

TL;DR

This paper introduces a machine learning framework using neural networks to improve the robustness and accuracy of reduced order models for nonlinear dynamical systems, demonstrated on the viscous Burgers equation.

Contribution

It proposes a novel neural network-based closure model trained with Bayesian regularization and extreme learning machine approaches for nonlinear model order reduction.

Findings

Significant accuracy improvements over standard Galerkin models

Models are computationally efficient and robust to parameter changes

Effective basis function selection enhances model performance

Abstract

Many reduced order models are neither robust with respect to the parameter changes nor cost-effective enough for handling the nonlinear dependence of complex dynamical systems. In this study, we put forth a robust machine learning framework for projection based reduced order modeling of such nonlinear and nonstationary systems. As a demonstration, we focus on a nonlinear advection-diffusion system given by the viscous Burgers equation, which is a prototype setting of more realistic fluid dynamics applications with the same quadratic nonlinearity. In our proposed methodology the effects of truncated modes are modeled using a singe layer feed-forward neural network architecture. The neural network architecture is trained by utilizing both the Bayesian regularization and extreme learning machine approaches, where the latter one is found to be computationally more efficient. A particular effort is devoted to the selection of basis functions considering the proper orthogonal decomposition and Fourier bases. It is shown that the proposed models yield significant improvements in the accuracy over the standard Galerkin projection models with a negligibly small computational overhead and provide reliable predictions with respect to the parameter changes.