Volume 56, pp. 52-65, 2022.

A comparison of reduced-order modeling approaches using artificial neural networks for PDEs with bifurcating solutions

Martin W. Hess, Annalisa Quaini, and Gianluigi Rozza

Abstract

This paper focuses on reduced-order models (ROMs) built for the efficient treatment of PDEs having solutions that bifurcate as the values of multiple input parameters change. First, we consider a method called local ROM that uses k-means algorithm to cluster snapshots and construct local POD bases, one for each cluster. We investigate one key ingredient of this approach: the local basis selection criterion. Several criteria are compared and it is found that a criterion based on a regression artificial neural network (ANN) provides the most accurate results for a channel flow problem exhibiting a supercritical pitchfork bifurcation. The same benchmark test is then used to compare the local ROM approach with the regression ANN selection criterion to an established global projection-based ROM and a recently proposed ANN based method called POD-NN. We show that our local ROM approach gains more than an order of magnitude in accuracy over the global projection-based ROM. However, the POD-NN provides consistently more accurate approximations than the local projection-based ROM.

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Key words

Navier–Stokes equations, reduced-order methods, reduced basis methods, parametric geometries, symmetry breaking bifurcation

AMS subject classifications

65P30, 35B32, 35Q30, 65N30, 65N35, 65N99

ETNA articles which cite this article

Vol. 56 (2022), pp. 235-255 Matthias Eichinger, Alexander Heinlein, and Axel Klawonn: Surrogate convolutional neural network models for steady computational fluid dynamics simulations

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