Asymptotically unbiased estimation of physical observables with neural samplers

We propose a general framework for the estimation of observables with generative neural samplers focusing on modern deep generative neural networks that provide an exact sampling probability. In this framework, we present asymptotically unbiased estimators for generic observables, including those that explicitly depend on the partition function such as free energy or entropy, and derive corresponding variance estimators. We demonstrate their practical applicability by numerical experiments for the two-dimensional Ising model which highlight the superiority over existing methods. Our approach greatly enhances the applicability of generative neural samplers to real-world physical systems.

Figure 1

Estimates for various observables around ${\beta }_c$. Neural Markov-chain Monte Carlo and neural importance sampling agree with the reference values provided by the exact analytical solutions as well as the Wolff algorithm. VAN deviates significantly. Observables are internal energy $U={\mathbb{E}}_p\left[H\right]$, absolute magnetization $\left|M\right|={\sum }_i{\mathbb{E}}_p\left(\right|s_i\left|\right)$, the free energy $F=\frac{-1}{\beta }ln\left(Z\right)$, and the entropy $S=-{\mathbb{E}}_p[lnp]$.

Figure 2

Zoom around critical ${\beta }_c\approx 0.4407$ showing the internal energy per lattice site from Fig. 1 (e.g., $16\times 16$ lattice). We took internal energy as a reference example, same considerations hold for the other observables. The estimates from our proposed method match with the exact reference. VAN is not shown because it is out of range as one can see from Fig. 1.

Figure 3

Estimation of observables during training for a single run on a $16\times 16$ lattice. The modified sampling procedure leads to accurate predictions at significantly earlier stages of training since it correct for imperfect samplers. As before, we look at the internal energy $U={\mathbb{E}}_p\left[H\right]$, the absolute magnetization $\left|M\right|={\sum }_i{\mathbb{E}}_p\left(\right|s_i\left|\right)$, the free energy $F=\frac{-1}{\beta }ln\left(Z\right)$, and the entropy $S=-{\mathbb{E}}_p[lnp]$.

Figure 4

Evolution of the acceptance rate (right) and the integrated autocorrelation time of the internal energy ${\tau }_{\mathrm{int},U}$ (left) during training. NMCMC runs were preformed on a $16\times 16$ lattice at ${\beta }_c$.

Figure 5

Samplers $q$ are trained at increasingly lower $\beta$ values and used to predict the internal energy $U/L^2$ at the critical coupling ${\beta }_c$. All results agree with the reference values within error bars. The variance of the estimators increase as the difference between model and target temperature gets larger. Transfer runs for NMCMC and NIS allow to only train one model which leads to significant speed up since runtime is dominated by training, as illustrated in the inset.

Figure 6

Histogram for the magnetization of the system at $\beta =0.55$. While the Metropolis algorithm is only able to capture one of the two modes of the distribution, NMCMC is able to cover both.