AI 中文总结
本文提出激活选择网络（ASN），将其应用于零量子核磁共振相关的自旋链哈密顿量本征值符号回归，可提取符合预期结构的表达式，且自适应激活选择无内在精度优势。
AI 中文摘要
参数依赖哈密顿量本征值的解析近似可提供仅通过数值对角化不易获得的物理见解。本文提出激活选择网络（ASN），这是一种可微分的符号回归架构，其中每个输入节点学习预定义解析函数的稀疏组合，训练后的网络可直接转换为显式表达式。回归前，哈密顿量参数和本征值以无量纲比率表示，该归一化确保量纲齐次性、减少自变量数量，并保证提取的表达式不依赖能量单位的选择。使用库{0, x, x²}时，连续隐藏层间的组合会生成次数逐渐升高的多项式展开，原则上通过增加网络深度可获得任意有限次数的多项式展开。将ASN应用于与零量子核磁共振相关的有效三位点和四位点自旋链哈密顿量。与简并微扰理论的比较显示，提取的表达式可捕获预期的常数、线性和二次结构。当预先指定合适的二次基时，固定基最小二乘模型的性能与ASN相当或略优，而加入径向特征可改善简并附近的局部近似。这些结果表明，ASN是一种可微分框架，可在存在多种合理函数形式时选择紧凑的符号表示，同时显示自适应激活选择相比合适的预定义基不具备内在精度优势。
英文摘要
Analytical approximations to eigenvalues of parameter-dependent Hamiltonians can provide physical insight that is not readily apparent from numerical diagonalization alone. Here, we introduce an activation-selection network (ASN), a differentiable symbolic-regression architecture in which each input node learns a sparse combination of predefined analytical functions, and the trained network can be converted directly into an explicit expression. Before regression, the Hamiltonian parameters and eigenvalues are expressed as dimensionless ratios. This normalization enforces dimensional homogeneity, reduces the number of independent variables, and ensures that the extracted expressions do not depend on the choice of energy units. Using the library {0, x, x^2}, compositions across successive hidden layers generate polynomial expansions of progressively higher degree; polynomial expansions of arbitrary finite degree can therefore be obtained in principle by increasing the network depth. We apply the ASN to effective three- and four-site spin-chain Hamiltonians relevant to zero-quantum nuclear magnetic resonance. Comparisons with degenerate perturbation theory show that the extracted expressions capture the expected constant, linear, and quadratic structure. Fixed-basis least-squares models match or slightly outperform the ASN when an appropriate quadratic basis is specified in advance, while inclusion of a radial feature improves the local approximation near the degeneracy. These results establish the ASN as a differentiable framework for selecting compact symbolic representations when several functional forms are plausible, while showing that adaptive activation selection does not provide an intrinsic accuracy advantage over a suitable predefined basis.
