Non-intrusive reduced order modeling of nonlinear problems using neural networks

•A novel non-intrusive RB method coupling proper orthogonal decomposition with multi-layer perceptrons.•Numerical studies for the nonlinear Poisson equation and the incompressible steady Navier–Stokes equations.•Physical and geometrical parametrizations considered.•Good agreement with the finite ele...

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Bibliographic Details
Published in:Journal of computational physics Vol. 363; pp. 55 - 78
Main Authors: Hesthaven, J.S., Ubbiali, S.
Format: Journal Article
Language:English
Published: Cambridge Elsevier Inc 15.06.2018
Elsevier Science Ltd
Subjects:
Artificial neural networks
Computational fluid dynamics
Computational physics
Decoupling
Dependence
Driven cavity flow
Fluid flow
Hypercubes
Incompressible flow
Latin hypercube sampling
Levenberg–Marquardt algorithm
Mathematical models
Multi-layer perceptron
Multilayers
Navier-Stokes equations
Neural networks
Non-intrusive reduced basis method
Nonlinear equations
Partial differential equations
Poisson distribution
Poisson equation
Proper Orthogonal Decomposition
Reduced order models
Poisson equation
Proper orthogonal decomposition
Levenberg–Marquardt algorithm
Driven cavity flow
Multi-layer perceptron
Non-intrusive reduced basis method
ISSN:0021-9991, 1090-2716
Online Access:Get full text
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Summary:•A novel non-intrusive RB method coupling proper orthogonal decomposition with multi-layer perceptrons.•Numerical studies for the nonlinear Poisson equation and the incompressible steady Navier–Stokes equations.•Physical and geometrical parametrizations considered.•Good agreement with the finite element method.•Substantial speed-up enabled at the online stage as compared to a traditional RB strategy.•Sensitivity analysis on the number of training samples and neurons.•Comparison between MLP-based regression and cubic spline interpolation within the RB framework. We develop a non-intrusive reduced basis (RB) method for parametrized steady-state partial differential equations (PDEs). The method extracts a reduced basis from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD) and employs artificial neural networks (ANNs), particularly multi-layer perceptrons (MLPs), to accurately approximate the coefficients of the reduced model. The search for the optimal number of neurons and the minimum amount of training samples to avoid overfitting is carried out in the offline phase through an automatic routine, relying upon a joint use of the Latin hypercube sampling (LHS) and the Levenberg–Marquardt (LM) training algorithm. This guarantees a complete offline-online decoupling, leading to an efficient RB method – referred to as POD-NN – suitable also for general nonlinear problems with a non-affine parametric dependence. Numerical studies are presented for the nonlinear Poisson equation and for driven cavity viscous flows, modeled through the steady incompressible Navier–Stokes equations. Both physical and geometrical parametrizations are considered. Several results confirm the accuracy of the POD-NN method and show the substantial speed-up enabled at the online stage as compared to a traditional RB strategy.
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ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2018.02.037