  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.
