波函数方差优化的被遗忘历史及其与神经网络变分蒙特卡洛的相关性

中文摘要 AI 辅助

我们重新审视了近一个世纪以来关于哪个局域能量泛函能最优地优化试探波函数的问题，这一问题在变分蒙特卡洛（VMC）以及更近期的神经网络变分蒙特卡洛（NN-VMC）中具有核心重要性。虽然方差优化可追溯至20世纪30年代，但现代神经网络波函数固有的高统计噪声和重尾局域能量分布重新激发了人们对这一方法的兴趣。我们在此追溯其漫长且 largely 被遗忘的历史，展示其与现代神经量子态（NQS）框架的直接相关性。最小化方差（一种$L^2$范数）隐含地假设局域能量分布为高斯分布：这是一个不合理的假设。对于库仑系统，局域能量分布表现出$E^{-4}$幂律尾部，导致中心极限定理对方差估计量失效。这种不稳定性可以通过稳健的代价函数来缓解：平均绝对偏差（MAD，一种$L^1$范数）、柯西损失或$L_{-4}$泛函，后者具有解析设计的尾部以匹配$E^{-4}$指数。我们在$H_2^+$上对这些泛函进行基准测试，$H_2^+$是一个在每个核间距下都可精确求解的系统，使用Guillemin-Zener波函数覆盖整个势能曲线。虽然能量最小化按构造产生最低能量，但方差最小化在每个$R$处都被其他泛函超越：MAD在成键区域表现更优，而$L_{-4}$在解离区域表现最佳。

英文摘要

We revisit the almost century-old question of which functional of the local energy best optimizes a trial wave function, a problem of central importance in Variational Monte Carlo (VMC) and, more recently, in Neural-Network VMC (NN-VMC). While variance optimization dates back to the 1930s, the high statistical noise and heavy-tailed local energy distributions inherent to modern neural-network wave functions have renewed interest in this approach. We retrace its long and largely forgotten history here, showing its direct relevance to modern Neural Quantum States (NQS) frameworks. Minimizing the variance (an $L^2$ norm) implicitly assumes a Gaussian local energy distribution: an unjustified assumption. For Coulombic systems, the local energy distribution exhibits $E^{-4}$ power-law tails, causing the Central Limit Theorem to fail for the variance estimator. This instability can be mitigated by robust cost functions: the Mean Absolute Deviation (MAD, an $L^1$ norm), the Cauchy loss, or the $L_{-4}$ functional, which features a tail analytically designed to match the $E^{-4}$ exponent. We benchmark these functionals on $H_2^+$, an exactly solvable system at every internuclear distance, using the Guillemin-Zener wave function across the full potential energy curve. While energy minimization by construction yields the lowest energy, variance minimization is surpassed at every $R$ by alternative functionals: MAD proves superior in the bonding region, while $L_{-4}$ performs best in the dissociation regime.

发表机构

• Università dell’Insubria（因苏布里亚大学）

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