Leveraging neural control variates for enhanced precision in lattice field theory

TL;DR

The paper tackles high variance in lattice field theory observables by introducing neural-network-parametrized control variates. It defines $f(\phi)=\sum_x\left(\frac{\partial g[\phi]_x}{\partial \phi_x}-g[\phi]_x\frac{\partial S}{\partial \phi_x}\right)$ with $\langle f\rangle=0$ and enforces translational invariance via $g_0$; training minimizes the empirical variance and uses a moving mean $\mu$ to prevent overfitting. In a 1+1D $\phi^4$ theory, the approach yields substantial variance reductions, with linear CVs sufficient at weak coupling but deeper networks needed at strong coupling, and transfer learning across time slices further boosts efficiency. The work shows promise for applying neural CVs to more complex theories, including gauge theories, while highlighting challenges in enforcing gauge invariance and achieving efficient training for 4D applications.

Abstract

Results obtained with stochastic methods have an inherent uncertainty due to the finite number of samples that can be achieved in practice. In lattice QCD this problem is particularly salient in some observables like, for instance, observables involving one or more baryons and it is the main problem preventing the calculation of nuclear forces from first principles. The method of control variables has been used extensively in statistics and it amounts to computing the expectation value of the difference between the observable of interest and another observable whose average is known to be zero but is correlated with the observable of interest. Recently, control variates methods emerged as a promising solution in the context of lattice field theories. In our current study, instead of relying on an educated guess to determine the control variate, we utilize a neural network to parametrize this function. Using 1+1 dimensional scalar field theory as a testbed, we demonstrate that this neural network approach yields substantial improvements. Notably, our findings indicate that the neural network ansatz is particularly effective in the strong coupling regime.

Figures (3)

• Figure 1: Training histories of the small and large couplings on $20 \times 20$ lattice. The figure shows the improvement of standard deviation using control variates with respect to the raw one $(\sigma_{\rm Raw}/\sigma_{\rm CV})$. The dashed line represents the zero hidden layer result (linear transformation), and the solid line represents the result with hidden layers. Networks for small and large couplings have 5 hidden layers and each has 4 neurons. For left and right panels, $10^4$ and $10^3$ samples are reserved for training the network respectively, and $10^3$ samples are used to estimate the variance.
• Figure 2: Training histories of control variates at $t=L_0/2=20$ on $40\times40$ lattice. The left plot shows the improvement of standard deviation with hidden layers. The right panel displays the training histories of different depths of networks with the large coupling, $\lambda=24.0$. $10^3$ samples are used to train the neural control variates and $10^3$ samples are employed to estimate the standard deviation.
• Figure 3: Correlation functions with $m^2=0.01$ and $\lambda=0.1$ on $40 \times 10$ lattice. The raw result and the result with control variates are shown. $2\times10^3$ samples are used in total and for the control variate result, $10^3$ samples are used for training and the whole samples are used for estimating observables. For the raw result with large sets, the correlators are calculated with $2 \times 10^5$ samples. The left plot shows the correlation functions with their fitting. Results are shifted horizontally for better visualization. The right plot displays the errors of the correlators in the left plot.