这篇论文用神经网络直接解量子场论方程，不需要标签数据，结果精度在百分级别，对初学者和专家都有启发。
论文用神经网络表示求解四维朗道规范杨-米尔斯理论的耦合鬼和胶子Dyson-Schwinger方程，仅从重整化方程残差训练得到解。与不动点解的误差在百分之几以内，且对初始化、网络大小、积分网格和红外边界条件的变化保持稳定。三胶子顶点模型变化产生的效应远大于神经网络误差。在截断限制内，复现了MiniMOM紫外跑动和胶子Schwinger函数的符号改变。
Neural solutions of coupled ghost and gluon Dyson--Schwinger equations in Landau gauge
The coupled ghost and gluon Dyson--Schwinger equations (DSEs) of four-dimensional Landau-gauge Yang--Mills (YM) theory are solved with a neural representation trained only from renormalized equation residuals. The neural and fixed-point solutions agree at the percent level and remain stable under changes of initialization, network size, integration grid, and infrared boundary condition. Variations of the three-gluon vertex model produce substantially larger effects than the neural error. The MiniMOM ultraviolet running and the sign change of the gluon Schwinger function are also reproduced within the limitations of the truncation.