用于精确少体计算的贝叶斯变分方法
Bayesian Variational Method for Precision Few-Body Calculations
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中文总结 AI 辅助
该研究针对量子多体系统变分描述受基函数数量限制的问题，提出贝叶斯变分方法（BVM），通过贝叶斯优化、增量对角化和修剪过程选择基函数，应用于GEM，大幅减少内存需求，为少体系统精确研究开辟新路径。
中文摘要 AI 辅助
许多量子多体系统的变分描述基于基函数展开，其实际限制常由所需基函数数量决定。我们提出贝叶斯变分方法（BVM），通过贝叶斯优化选择基函数：高斯过程代理模型根据已评估候选者预测最可能降低能量的候选者，且候选者评估相互独立分布在多个节点。增量对角化通过重用已接受基的先前对角化来评估候选者，修剪过程持续去除几乎线性相关的基函数。BVM广泛适用于基于量子力学中基函数展开的能量变分问题，本文将其应用于少体物理中的标准方法高斯展开方法（GEM）。由于GEM基是非正交的，线性相关性强，BVM实现的基约简大，既加速计算，更重要的是大幅降低内存需求。通过仅705个基函数可将完整32000维GEM对角化的参考能量精确到0.01K以内，2127个基函数时精确到0.001K以内，内存减少99.95%和99.56%。在GEM内，这为六体和七体系统及更复杂系统的精确研究开辟了道路。
英文摘要
Many variational descriptions of quantum many-body systems rest on an expansion over basis functions, and their practical limit is often set by the number of basis functions required. We propose the Bayesian variational method (BVM), in which the basis functions are selected by Bayesian optimization: a Gaussian-process surrogate model, conditioned on the candidates evaluated so far, predicts which candidates are most likely to lower the energy, and the candidate evaluations, being mutually independent, are distributed over many nodes. Two further ingredients make the method practical. An incremental diagonalization evaluates each candidate by reusing the previous diagonalization of the accepted basis instead of solving the full eigenvalue problem anew. A trimming procedure continually removes basis functions that have become nearly linearly dependent, keeping the accepted basis small while guiding it toward the optimal solution. The BVM applies broadly to energy variational problems based on basis-function expansions in quantum mechanics; here we apply it to the Gaussian expansion method (GEM), a standard approach in few-body physics. Because the GEM basis is nonorthogonal, its linear dependence is strong, so the basis reduction achieved by the BVM is large. The reduction both accelerates the computation and, more importantly, greatly reduces the memory requirement, one of the central bottlenecks of the variational method: the reference energy of the full 32,000-dimensional GEM diagonalization is reproduced to within 0.01 K with only 705 basis functions and to within 0.001 K with 2,127, corresponding to memory reductions of 99.95% and 99.56%, since the matrix storage grows as the square of the basis dimension. Within the GEM, this opens a path to the precision study of six- and seven-body systems, and beyond, that has so far been difficult to reach.
