A Comparison of Reduced-Order Modeling Approaches Using Artificial Neural Networks for PDEs with Bifurcating Solutions

TL;DR

This paper compares various reduced-order modeling techniques for PDEs with bifurcating solutions, highlighting the effectiveness of local ROM with ANN-based basis selection and contrasting it with global ROM and POD-NN methods.

Contribution

It introduces a local ROM approach with ANN-based basis selection and compares its performance against global ROM and POD-NN for bifurcating PDE solutions.

Findings

Local ROM with ANN basis selection outperforms global projection ROM in accuracy.

POD-NN consistently surpasses local ROM in approximation accuracy.

The ANN-based local basis criterion is most effective for bifurcating solutions.

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.