Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general eigenvalue problem, which can be transformed into an optimization problem with an objective function given by a generalized Rayleigh quotient. The proposed algorithm requires iterative computations. However, it enables efficient annealing-based computation of eigenvectors to arbitrary precision without increasing the number of variables. Investigations using simulated annealing demonstrate how the number of iterations scales with system size and annealing time. Computational performance depends on system size, annealing time, and problem characteristics.

翻译：采用基于退火的方法求解偏微分方程（PDEs）涉及求解广义特征值问题。对PDE进行离散化会得到一个线性方程组（SLE）。求解SLE可表述为一个广义特征值问题，该问题可转化为以广义瑞利商为目标函数的优化问题。所提出的算法需要迭代计算。然而，它能够在变量数量不增加的情况下，通过基于退火的高效计算获得任意精度的特征向量。利用模拟退火的研究展示了迭代次数如何随系统规模和退火时间变化。计算性能取决于系统规模、退火时间及问题特性。