Solving partial differential equations using an annealing-based approach is based on solving generalized eigenvalue problems. When a partial differential equation is discretized, it leads to a system of linear equations (SLE). Solving an SLE can be expressed as a general eigenvalue problem, which can be converted into an optimization problem with the objective function being a generalized Rayleigh quotient. The proposed algorithm allows the computation of eigenvectors at arbitrary precision without increasing the number of variables using an Ising machine. Simple examples solved using this method and theoretical analysis provide a guideline for appropriate parameter settings.

翻译：采用退火方法求解偏微分方程的核心在于求解广义特征值问题。当偏微分方程被离散化后，会转化为线性方程组（SLE）的求解问题。求解SLE可表述为广义特征值问题，进而可转化为以广义瑞利商为目标函数的优化问题。所提出的算法利用伊辛机，能够在变量数量不增加的情况下，以任意精度计算特征向量。通过该方法求解的简单算例及理论分析，为参数设置提供了指导原则。