用量子传播子思想优化支持向量机核函数，提升物理系统回归精度。
Support Vector Machine Kernels as Quantum Propagators
- 将核函数与量子传播子对应，依据格林函数谱对齐选择最优核
- 在电导、能带等系统中，新方法显著优于传统核函数
- 适合需物理先验的复杂系统建模，如凝聚态与光子晶体
物理系统回归中的核函数选择仍依赖试错，缺乏理论指导。本文建立支持向量机核函数与量子传播子之间的数学对应关系，证明核函数性能取决于其谱分布与系统格林函数的对齐程度。基于此同构性，提出统一的物理引导核设计框架：对已知传播子形式的系统，可推导出标准核到物理算符的映射规则；对格林函数解析不可解的复杂系统，提出基于核多项式法与杰克逊平滑的数值构造方法，生成定制化物理对齐核。在电导、电子能带、非谐振子和光子晶体等多个系统上的数值实验表明，只要存在格林函数对齐，该框架始终表现优异。
原文摘要 · Abstract (English)
Selecting optimal kernels for regression in physical systems remains a challenge, often relying on trial-and-error with standard functions. In this work, we establish a mathematical correspondence between support vector machine kernels and quantum propagators, demonstrating that kernel efficacy is determined by its spectral alignment with the system's Green's function. Based on this isomorphism, we propose a unified, physics-informed framework for kernel selection and design. For systems with known propagator forms, we derive analytical selection rules that map standard kernels to physical operators. For complex systems where the Green's function is analytically intractable, we introduce a constructive numerical method using the Kernel Polynomial Method with Jackson smoothing to generate custom, physics-aligned kernels. Numerical experiments spanning electrical conductivity, electronic band structure, anharmonic oscillators, and photonic crystals demonstrate that this framework consistently performs well as long as there is an alignment with a Green's function.
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