Flow-based generative models for Markov chain Monte Carlo in lattice field theory (1904.12072v3)

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 $\phi^4$ theory in two dimensions.

• The paper introduces a flow-based MCMC algorithm that uses neural network-parameterized normalizing flows to efficiently sample lattice field theory configurations.
• The study benchmarks the method against HMC and local Metropolis, achieving up to 70% acceptance rates and significantly reduced autocorrelation times.
• The approach mitigates critical slowing down and enables parallelized sampling for large-scale lattice simulations, opening avenues for broader applications.

Overview of Flow-Based Generative Models for MCMC in Lattice Field Theory

The paper introduces a novel approach to Markov chain Monte Carlo (MCMC) simulations in lattice field theory using flow-based generative models. The authors propose a machine learning paradigm that leverages the expressiveness of normalizing flows to construct an efficient MCMC algorithm. This method addresses the challenge of evaluating path integrals in lattice field theories by generating Markov chains with reduced autocorrelation times, even in regimes where traditional techniques face critical slowing down.

The proposed algorithm fundamentally combines flow-based techniques with the Metropolis-Hastings framework to maintain exactness in sampling. The flow model is trained to approximate the target distribution, thus allowing the MCMC process to produce samples with improved decorrelation properties. The flow model's flexibility enables sampling without pre-existing configurations from the target distribution, which is a distinct advantage over conventional methods.

Technical Contributions

1. Flow-Based MCMC Algorithm: The authors develop a flow-based generative model specifically for MCMC sampling. By parameterizing distributions with neural networks, the model optimizes a variational family to approximate the target distribution. This approach reduces autocorrelation and mitigates issues such as critical slowing down by leveraging efficient change-of-variable transformations.
2. Comparison with HMC and Local Metropolis: The algorithm's performance is benchmarked against Hybrid Monte Carlo (HMC) and local Metropolis algorithms within the context of 2D $\phi^4$ theory. The paper demonstrates that the flow-based approach significantly improves acceptance rates and reduces correlation length, suggesting its potential to supersede traditional algorithms amidst critical phenomena.
3. Autocorrelation and Acceptance Analysis: The paper defines an observable-independent estimator for autocorrelation time based on acceptance statistics. This provides insights into how the acceptance rate directly correlates with reduced autocorrelation, offering a theoretical underpinning for the empirical success of the flow-based method.

Numerical Results

The empirical results presented confirm the viability of the flow-based MCMC algorithm in lattice field theory applications. The method was tested on $\phi^4$ theory, where it showed competitive results with significantly enhanced acceptance rates (achieving up to 70%) and reduced autocorrelation times across various lattice sizes. This suggests that the approach is robust against critical slowing down.

Implications and Future Directions

The implications of this research extend to broader applications in lattice field theories, including complex domains like quantum chromodynamics (QCD). The elimination of critical slowing down in the sampling process offers promising efficiency gains for large-scale lattice simulations. Notably, the independence of proposed samples means parallelization is feasible, enhancing computational efficiency further.

This research opens a pathway for future explorations into optimizing training costs and scaling for more intricate lattice systems. The paper hints at potential directions, including leveraging convolutional structures to improve scaling and incorporating physical symmetries to enhance model efficiency.

The adoption of flow-based methods in lattice field theory could revolutionize how numerical simulations are approached, especially in contexts requiring immense computational resources. Future work might include incorporating gauged symmetries and exploring manifold-constrained flows that better capture the structure inherent in gauge theories.