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General-purpose Markov Chain Monte Carlo sampling algorithms suffer from adramatic reduction in efficiency as the system being studied is driven towardsa critical point. Recently, a series of seminal studies suggested thatnormalizing flows - a class of deep generative models - can form the basis of asampling strategy that does not suffer from this 'critical slowing down'. Thecentral idea is to use machine learning techniques to build (approximate)trivializing maps, i.e. field transformations that map the theory of interestinto a 'simpler' theory in which the degrees of freedom decouple, and where thestatistical weight in the path integral is given by a distribution from whichsampling is easy. No separate process is required to generate training data forsuch models, and convergence to the desired distribution is guaranteed througha reweighting procedure such as a Metropolis test. In a proof-of-principledemonstration on two-dimensional $\phi^4$ theory, Albergo et al.(arXiv:1904.12072) modelled the trivializing map as a sequence of pointwiseaffine transformations. We pick up this thread, with the aim of quantifying howwell we can expect this approach to scale as we increase the number of degreesof freedom in the system. We make several modifications to the original designthat allow our models learn more efficient representations of trivializing mapsusing much smaller neural networks, which leads to a large reduction in thecomputational cost required to train models of equivalent quality. After makingthese changes, we find that sampling efficiency is almost entirely dictated byhow extensively a model has been trained, while being unresponsive to furtheralterations that increase model flexibility. However, as we move towards thecontinuum limit the training costs scale extremely quickly, which urgentlyrequires further work to fully understand and mitigate.

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