让神经网络保持对称性，提升格点规范场理论的计算精度
Symmetry-preserving neural networks in lattice field theories
- 设计保持规范对称性的卷积网络（L-CNN）
- 在威尔逊环回归任务中表现优于传统网络
- 适合研究格点量子场论与对称性相关的物理问题
本论文研究具有对称性保持特性的神经网络，并展示其在格点场论中的优势。通过解释等变性概念，阐明其对保持对称性的关键作用。首先在复标量场模型中验证平移对称性带来的好处；随后针对规范理论，专门设计了格点规范等变卷积神经网络（L-CNN）。L-CNN成功解决了威尔逊环等物理可观测量的回归问题，而缺乏规范对称性的传统架构则表现显著较差。最后提出神经梯度流技术——由神经网络求解常微分方程，用于生成格点规范场构型。
原文摘要 · Abstract (English)
This thesis deals with neural networks that respect symmetries and presents the advantages in applying them to lattice field theory problems. The concept of equivariance is explained, together with the reason why such a property is crucial for the network to preserve the desired symmetry. The benefits of choosing equivariant networks are first illustrated for translational symmetry on a complex scalar field toy model. The discussion is then extended to gauge theories, for which Lattice Gauge Equivariant Convolutional Neural Networks (L-CNNs) are specifically designed ad hoc. Regressions of physical observables such as Wilson loops are successfully solved by L-CNNs, whereas traditional architectures which are not gauge symmetric perform significantly worse. Finally, we introduce the technique of neural gradient flow, which is an ordinary differential equation solved by neural networks, and propose it as a method to generate lattice gauge configurations.
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