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Advances in Computational Mathematics

, Volume 44, Issue 6, pp 1717–1750 | Cite as

Neural network closures for nonlinear model order reduction

  • Omer San
  • Romit Maulik
Article

Abstract

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.

Keywords

Machine learning Neural networks Extreme learning machine Model order reduction Projection methods Burgers equation 

Mathematics Subject Classification (2010)

37N10 76M25 76F20 76D99 

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Notes

Acknowledgments

The computing for this project was performed by using resources from the High Performance Computing Center (HPCC) at Oklahoma State University.

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Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringOklahoma State UniversityStillwaterUSA

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