  General-purpose Markov Chain Monte Carlo sampling algorithms suffer from a
dramatic reduction in efficiency as the system being studied is driven towards
a critical point. Recently, a series of seminal studies suggested that
normalizing flows - a class of deep generative models - can form the basis of a
sampling strategy that does not suffer from this 'critical slowing down'. The
central idea is to use machine learning techniques to build (approximate)
trivializing maps, i.e. field transformations that map the theory of interest
into a 'simpler' theory in which the degrees of freedom decouple, and where the
statistical weight in the path integral is given by a distribution from which
sampling is easy. No separate process is required to generate training data for
such models, and convergence to the desired distribution is guaranteed through
a reweighting procedure such as a Metropolis test. In a proof-of-principle
demonstration on two-dimensional $\phi^4$ theory, Albergo et al.
(arXiv:1904.12072) modelled the trivializing map as a sequence of pointwise
affine transformations. We pick up this thread, with the aim of quantifying how
well we can expect this approach to scale as we increase the number of degrees
of freedom in the system. We make several modifications to the original design
that allow our models learn more efficient representations of trivializing maps
using much smaller neural networks, which leads to a large reduction in the
computational cost required to train models of equivalent quality. After making
these changes, we find that sampling efficiency is almost entirely dictated by
how extensively a model has been trained, while being unresponsive to further
alterations that increase model flexibility. However, as we move towards the
continuum limit the training costs scale extremely quickly, which urgently
requires further work to fully understand and mitigate.
