利用马尔可夫链生成的关联样本训练神经控制变量，提升蒙特卡洛模拟效率
Training neural control variates using correlated configurations
- 用相关采样数据训练神经网络学习辅助函数
- 在计算资源有限时显著降低估计方差
- 适用于高维物理模拟中的高效采样设计
神经控制变量（NCVs）是蒙特卡洛（MC）模拟中降低方差的强大工具，尤其在传统控制变量难以解析构造的高维问题中。通过训练神经网络学习与目标可观测量相关的辅助函数，NCVs可在保持无偏性的同时显著减少估计方差。然而，一个常被忽视的关键问题是：马尔可夫链蒙特卡洛（MCMC）生成的自相关样本通常因统计冗余被丢弃用于误差估计，但其可能包含有助于训练的分布结构信息。本文系统研究了在训练神经控制变量时使用这些相关配置的影响。我们从概念和数值上证明，利用相关数据可提升控制变量性能，特别是在计算资源受限的情况下。实验基于U(1)规范理论和标量场理论，展示了何时及如何通过相关样本增强NCV构建。结果为高效利用MCMC数据训练神经网络提供了实用指导。
原文摘要 · Abstract (English)
Neural control variates (NCVs) have emerged as a powerful tool for variance reduction in Monte Carlo (MC) simulations, particularly in high-dimensional problems where traditional control variates are difficult to construct analytically. By training neural networks to learn auxiliary functions correlated with the target observable, NCVs can significantly reduce estimator variance while preserving unbiasedness. However, a critical but often overlooked aspect of NCV training is the role of autocorrelated samples generated by Markov Chain Monte Carlo (MCMC). While such samples are typically discarded for error estimation due to their statistical redundancy, they may contain useful information about the structure of the underlying probability distribution that can benefit the training process. In this work, we systematically examine the effect of using correlated configurations in training neural control variates. We demonstrate, both conceptually and numerically, that training on correlated data can improve control variate performance, especially in settings with limited computational resources. Our analysis includes empirical results from $U(1)$ gauge theory and scalar field theory, illustrating when and how autocorrelated samples enhance NCV construction. These findings provide practical guidance for the efficient use of MCMC data in training neural networks.
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