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

Abstract

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 ${\varphi }^4$ 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.

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Cited by in corpus (25)

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• Stochastic normalizing flows as non-equilibrium transformations
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• Learning Lattice Quantum Field Theories with Equivariant Continuous Flows
• Learning Trivializing Gradient Flows for Lattice Gauge Theories
• Diffusion Models as Stochastic Quantization in Lattice Field Theory
• Flow-based density of states for complex actions
• Sampling the lattice Nambu-Goto string using Continuous Normalizing Flows
• Exploring phase space with Nested Sampling
• Parallel Tempered Metadynamics: Overcoming potential barriers without surfing or tunneling
• Adaptive Monte Carlo augmented with normalizing flows
• Scaling of Stochastic Normalizing Flows in $\mathrm{SU}(3)$ lattice gauge theory
• Flow-Based Sampling for Entanglement Entropy and the Machine Learning of Defects
• Simulating first-order phase transition with hierarchical autoregressive networks
• Machine-learned RG-improved gauge actions and classically perfect gradient flows
• Self-learning Monte Carlo with equivariant Transformer
• Non-Perturbative Trivializing Flows for Lattice Gauge Theories
• Flow-based Nonperturbative Simulation of First-order Phase Transitions
• Scaling flow-based approaches for topology sampling in $\mathrm{SU}(3)$ gauge theory
• Sampling metastable systems using collective variables and Jarzynski-Crooks paths
• Hierarchical autoregressive neural networks in three-dimensional statistical system
• Uncertainty in AI-driven Monte Carlo simulations
• Group-Equivariant Diffusion Models for Lattice Field Theory