AI 中文总结
该研究开发正张量网络参数化方法计算卡拉比-丘流形的里奇平坦凯勒度量，在gCICY三维流形测试中，其误差低于参数匹配神经势及同次无约束埃尔米特度量，且存在计算饱和特性。
AI 中文摘要
我们开发并实现了一种正张量网络参数化方法，用于计算卡拉比-丘流形上的里奇平坦凯勒度量。该方法用矩阵乘积分解取代了高次代数度量的大埃尔米特系数矩阵。对于此处使用的浸入源空间，所得度量在每个参数值下均为全局正的，且在固定局部维度和键维度时，其参数数量仅随代数度数线性增长。我们在三个广义完全相交卡拉比-丘（gCICY）三维流形上测试了该构造，逐图构造了定义并训练该度量所需的广义截面、全纯体积形式和采样测度。从一个公共低次度量出发，该张量网络在每轮配对运行中均优于参数匹配的神经势（使用相同截面数据），同时降低了体误差和1%尾部条件均值；在相同起始条件下，相较于直接优化同次无约束埃尔米特度量，其误差也显著更低，两种方法均在各自调度下优化至验证收敛。在第二种几何上，参数数量少于低次无约束埃尔米特基线的高次网络，大幅降低了相同样本的误差。我们进一步在测试计算中观察到饱和现象：在固定键维度下，提高度数最终会进入平台；在固定优化工作量下，提高键维度无已分辨增益；且结果强烈依赖于初始化和优化路径。
英文摘要
We develop and implement a positive tensor-network parameterization for computing Ricci-flat Kähler metrics on Calabi-Yau manifolds. It replaces the large Hermitian coefficient matrix of a high-degree algebraic metric by a matrix-product factorization. For the immersed source spaces used here, the resulting metric is globally positive for every parameter value and, at fixed local and bond dimensions, its number of parameters grows only linearly with the algebraic degree. We test the construction on three generalized complete-intersection Calabi-Yau (gCICY) threefolds, constructing chart by chart the generalized sections, holomorphic volume forms and sampling measures that define and train the metric there. From a common low-degree metric, the tensor network outperforms a parameter-matched neural potential using the same section data, reducing both bulk errors and the one-percent tail conditional mean in every paired run. It also reaches a substantially lower error than direct optimization of an unrestricted Hermitian metric of the same degree from the same start, with both methods optimized to validation convergence under their respective schedules. On a second geometry, a higher-degree network with fewer parameters than a lower-degree unrestricted Hermitian baseline substantially reduces the same-sample errors. We further observe saturation within the tested calculations: at fixed bond dimension, increasing the degree eventually plateaus; increasing the bond dimension at fixed optimization effort gives no resolved gain; and the outcome depends strongly on initialization and optimization path.
