基于机器学习的全息超导体性质研究
Properties of holographic superconductors from Machine Learning
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中文总结 AI 辅助
本研究采用机器学习启发的优化技术，结合两种试探函数与L-BFGS-B算法，研究全息超导体的临界温度，其数值结果与分析预测吻合，证实了该方法的稳健性与实用性。
中文摘要 AI 辅助
我们采用受机器学习启发的现代优化技术研究全息超导体，通过最小化本征值λ²的变分泛函来获取临界温度，使用两种互补的试探函数：简单的余弦近似F(z)=cos(a z)和灵活的指数多项式近似F(z)=exp(∑ₙ=2ᴺ aₙ zⁿ)，二者均自动满足标准边界条件。对于余弦近似，我们执行单参数最小化，在Δ的宽范围内得到λ²(Δ)和T_c/√ρ，其中包括Δ=1和Δ=2处的高精度精确值；指数多项式近似最多含19个系数，采用带热启动的多起点L-BFGS-B算法优化，与已知精确结果的吻合度更高。我们的λ²(Δ)和T_c/√ρ数值数据与文献中的分析预测一致，证实了变分方法的稳健性。因此，本研究表明，分析试探函数与现代数值优化的结合，为探索全息超导体提供了强大、灵活且高效的工具，可方便扩展至包含反作用或该领域其他部分的研究。
英文摘要
We investigate holographic superconductors using modern optimisation techniques inspired by machine learning. The critical temperature is obtained by minimising the variational functional for the eigenvalue $λ^2$ with two complementary trial functions: a simple cosine ansatz $F(z)=\cos(a z)$ and a flexible exponential polynomial $F(z)=\exp(\sum_{n=2}^{N} a_n z^n)$, both of which automatically satisfy the standard boundary conditions. For the cosine ansatz, we perform a one-parameter minimisation and obtain $λ^2(Δ)$ and $T_c/\sqrtρ$ over a wide range of $Δ$, including the exact values at $Δ=1$ and $Δ=2$ to high accuracy. The exponential polynomial ansatz, with up to 19 coefficients, is optimised using a multi-start L-BFGS-B algorithm with warm-starting, yielding even better agreement with known exact results. Our numerical data for $λ^2(Δ)$ and $T_c/\sqrtρ$ match the analytical predictions from the literature, confirming the robustness of the variational approach. This work; therefore, demonstrates that a combination of analytic trial functions and modern numerical optimisation provides a powerful, flexible, and efficient tool for exploring holographic superconductors, and can be readily extended to include backreaction or other sectors in this field.
