Estimation of Thermodynamic Observables in Lattice Field Theories with Deep Generative Models

In this Letter, we demonstrate that applying deep generative machine learning models for lattice field theory is a promising route for solving problems where Markov chain Monte Carlo (MCMC) methods are problematic. More specifically, we show that generative models can be used to estimate the absolute value of the free energy, which is in contrast to existing MCMC-based methods, which are limited to only estimate free energy differences. We demonstrate the effectiveness of the proposed method for two-dimensional ${\varphi }^4$ theory and compare it to MCMC-based methods in detailed numerical experiments.

Figure 1

Absolute magnetization density as a function of hopping parameter $\kappa$ for bare coupling $\lambda =0.022$. Results for various lattice sizes overlap. The values were estimated with an overrelaxed Hamiltonian Monte Carlo (HMC) [43, 44, 45, 46]. The dashed line denotes the hopping parameter value $\kappa =0.3$ for the free energy estimation in the numerical experiments.

Figure 2

Estimate of free energy density at $\lambda =0.022$ and $\kappa =0.3$ obtained by both the flow- and MCMC-based methods for various lattice sizes. MCMC estimates are obtained from integrating free energy differences. Both methods use the same number of samples (5.6 M) for estimation. Errors are obtained with the delta and uwerr method [48] for flow and HMC, respectively (see the Supplemental Material [27] for jackknife error analysis and Appendix of [48] for theory of the jackknife method).