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

Data-Driven Enhanced Model Reduction for Bifurcating Models in Computational Fluid Dynamics

“…This benchmark test has been adapted from various papers, see, e.g., [4,3,6,17] and the references cited therein. At the inlet, We prescribe a parabolic horizontal velocity component with maximum 9/4 and zero vertical component, as shown in [12], Fig. 3.1.…”

“…On the rest of the boundary, we impose a no-slip condition. Reference solutions are plotted in [12], Fig. 2.1 on a fine 40 × 41 uniform parameter grid.…”

“…3. Here, the model reduction approach presented in [12] will be extended to resolve multiple solution branches as shown in Fig. 4.…”

“…Transitions from steady solutions to time-periodic and finally fully time-dependent solutions without any discernible period occur as a parameter is varied in a domain containing Hopf bifurcations. ROMs for the jet flow problem have been previously studied in [12,8,11,20,10,14], while an investigation of ROMs for the cavity flow problem can be found in [13,8]. Various intrusive and also data-driven ROMs for bifurcating problems have been investigated also in [25,7,23,22,21,2,16,19,18,15].…”

“…In this paper, we discuss several data-driven enhancements to existing ROMs. We focus on two variants of the dynamic mode decomposition (DMD) algorithm, namely the Hankel-DMD and a stabilized DMD, to improve the ROM for the cavity flow problem introduced in [12]. For the jet flow problem, we propose to use artificial neural networks (ANNs) to assign a local ROM [8] to a given region of the parameter domain.…”