Efficient Modelling of Trivializing Maps for Lattice $\phi^4$ Theory Using Normalizing Flows: A First Look at Scalability.

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 element-wise 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, facilitating substantial improvements in the efficiency of the sampling stage. 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.