Flow-based generative models for Markov chain Monte Carlo in lattice field theory

Abstract

A Markov chain update scheme using a machine-learned flow-based generative model is proposed for Monte Carlo sampling in lattice field theories. The generative model may be optimized (trained) to produce samples from a distribution approximating the desired Boltzmann distribution determined by the lattice action of the theory being studied. Training the model systematically improves autocorrelation times in the Markov chain, even in regions of parameter space where standard Markov chain Monte Carlo algorithms exhibit critical slowing down in producing decorrelated updates. Moreover, the model may be trained without existing samples from the desired distribution. The algorithm is compared with HMC and local Metropolis sampling for φ 4 theory in two dimensions.

Equivariant Flow-Based Sampling for Lattice Gauge Theory

“…This is true, in particular, for the lattice formulation of quantum chromodynamics (QCD) [21][22][23], which enables calculations of nonperturbative phenomena arising from the standard model of particle physics. Recently, there has been progress in the development of flow-based generative models which can be trained to directly produce samples from a given probability distribution; early success has been demonstrated in theories of bosonic matter, spin systems, molecular systems, and for Brownian motion [24][25][26][27][28][29][30][31][32][33][34]. This progress builds on the great success of flow-based approaches for image, text, and structured object generation [35][36][37][38][39][40][41][42], as well as non-flow-based machine learning techniques applied to sampling for physics [43][44][45][46][47][48].…”

“…This feature enables training the flow model, i.e., optimizing the function f, by minimizing the distance between the model probability density qðU 0 Þ and the desired density pðU 0 Þ using a chosen metric. Any deviation from the true distribution due to an imperfect model can be corrected by a number of techniques; in this Letter, we apply independence Metropolis sampling [26]. (Reweighted observables can also be used [59,60].…”

“…After training, the flow-based models were used to generate proposals for an independence Metropolis Markov chain [26], producing ensembles of 100000 samples each. For comparison, ensembles of identical size were produced using the HMC and heat bath algorithms.…”

Quantum tunnelling, real-time dynamics and Picard-Lefschetz thimbles

“…This would mirror the efficiency gained for imaginary time lattice simulations through Hybrid Monte Carlo or Metropolis with local update. An interesting option is to implement a machine learning algorithm to better estimate the thimble [53,54]. Another is to perhaps adapt a complex Langevin algorithm [46,55,56] to explore along a thimble rather than in the entire complexified manifold.…”

Anomaly detection with density estimation

“…Other neural network approaches to density estimation have been studied in high energy physics. Such methods include Generative Adversarial Networks (GANs) , autoencoders [54,66], physically-inspired networks [67,68], and flows [69,70]. GANs are efficient for sampling from a density and are thus promising for accelerating slow simulations, but they do not provide an explicit representation of the density itself.…”