提出一种更稳定的对称化方法，提升神经网络求解多电子薛定谔方程的性能。
Diagonal Symmetrization of Neural Network Solvers for the Many-Electron Schrödinger Equation
- 训练后平均法比训练中对称化更稳定有效
- 在变分蒙特卡洛下，训练中对称化反而降低性能
- 适合需要高精度量子模拟的研究者
将群对称性融入神经网络是人工智能赋能科学的重要基础。对角群对称性（描述多个粒子同步运动下的不变性）自然出现在多体量子问题中。尽管重要，但因其缺乏通用的不变映射，研究较少。本文探讨了在变分蒙特卡洛训练的神经网络泛函中引入对角不变性的不同方法，包括数据增强、群平均和规范变换。结果表明，与标准机器学习相反，在训练中进行对称化会破坏稳定性并导致性能下降。理论与数值分析显示，这种异常行为可能源于标准机器学习中不存在的独特计算-统计权衡。相比之下，训练后平均法对这种权衡不敏感，成为一种简单、灵活且高效的方法。
原文摘要 · Abstract (English)
Incorporating group symmetries into neural networks has been a cornerstone of success in many AI-for-science applications. Diagonal groups of isometries, which describe the invariance under a simultaneous movement of multiple objects, arise naturally in many-body quantum problems. Despite their importance, diagonal groups have received relatively little attention, as they lack a natural choice of invariant maps except in special cases. We study different ways of incorporating diagonal invariance in neural network ansätze trained via variational Monte Carlo methods, and consider specifically data augmentation, group averaging and canonicalization. We show that, contrary to standard ML setups, in-training symmetrization destabilizes training and can lead to worse performance. Our theoretical and numerical results indicate that this unexpected behavior may arise from a unique computational-statistical tradeoff not found in standard ML analyses of symmetrization. Meanwhile, we demonstrate that post hoc averaging is less sensitive to such tradeoffs and emerges as a simple, flexible and effective method for improving neural network solvers.
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