Neural network closures for nonlinear model order reduction

• Published in: Advances in computational mathematics Vol. 44; no. 6; pp. 1717 - 1750
• Main Authors: San, Omer, Maulik, Romit
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2018; Springer Nature B.V
• Subjects: Basis functions; Bayesian analysis; Burgers equation; Closures; Computational fluid dynamics; Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Computer architecture; Dependence; Forecasting; Galerkin method; Machine learning; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Model accuracy; Model reduction; Neural networks; Nonlinear systems; Nonlinearity; Parameter robustness; Proper Orthogonal Decomposition; Reduced order models; Regularization; System effectiveness; Visualization; 37N10; 76M25; Extreme learning machine; Neural networks; 76F20; 76D99; Machine learning; Projection methods; Model order reduction; Burgers equation
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-018-9590-z

Many reduced-order models are neither robust with respect to 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 prototypical setting of more realistic fluid dynamics applications due to its quadratic nonlinearity. In our proposed methodology the effects of truncated modes are modeled using a single 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 more computationally efficient. A significant emphasis is laid on the selection of basis functions through the use of both Fourier bases and proper orthogonal decomposition. It is shown that the proposed model yields significant improvements in accuracy over the standard Galerkin projection methodology with a negligibly small computational overhead and provide reliable predictions with respect to parameter changes.