  The ability to extract generative parameters from high-dimensional fields of
data in an unsupervised manner is a highly desirable yet unrealized goal in
computational physics. This work explores the use of variational autoencoders
(VAEs) for non-linear dimension reduction with the specific aim of {\em
disentangling} the low-dimensional latent variables to identify independent
physical parameters that generated the data. A disentangled decomposition is
interpretable, and can be transferred to a variety of tasks including
generative modeling, design optimization, and probabilistic reduced order
modelling. A major emphasis of this work is to characterize disentanglement
using VAEs while minimally modifying the classic VAE loss function (i.e. the
Evidence Lower Bound) to maintain high reconstruction accuracy. The loss
landscape is characterized by over-regularized local minima which surround
desirable solutions. We illustrate comparisons between disentangled and
entangled representations by juxtaposing learned latent distributions and the
true generative factors in a model porous flow problem. Hierarchical priors are
shown to facilitate the learning of disentangled representations. The
regularization loss is unaffected by latent rotation when training with
rotationally-invariant priors, and thus learning non-rotationally-invariant
priors aids in capturing the properties of generative factors, improving
disentanglement. Finally, it is shown that semi-supervised learning -
accomplished by labeling a small number of samples ($O(1\%)$) - results in
accurate disentangled latent representations that can be consistently learned.
