Department of Experimental Ophthalmology, Saarland University, Homburg/Saar, Germany.
Dr. Rolf M. Schwiete Center for Limbal Stem Cell and Aniridia Research, Saarland University, Homburg/Saar, Germany.
Acta Ophthalmol. 2023 Nov;101(7):775-782. doi: 10.1111/aos.15664. Epub 2023 Mar 21.
To investigate particle swarm optimisation (PSO) as a modern purely data driven non-linear iterative strategy for lens formula constant optimisation in intraocular lens power calculation.
A PSO algorithm was implemented for optimising the root mean squared formula prediction error (rmsPE, defined as achieved refraction minus predicted refraction) for the Castrop formula in a dataset of N = 888 cataractous eyes with implantation of the Hoya Vivinex hydrophobic acrylic aspheric lens. The hyperparameters were set to inertia: 0.8, accelerations c1 = c2 = 0.1. The algorithm was initialised with N = 100 particles having random positions and velocities within the box constraints of the constant triplet parameter space C = 0.25 to 0.45, H = -0.25 to 0.25 and R = -0.25 to 0.25. The performance of the algorithm was compared to classical gradient-based Trust-Region-Reflective and Interior-Point algorithms.
The PSO algorithm showed fast and stable convergence after 37 iterations. The rmsPE reduced systematically to 0.3440 diopters (D). With further iterations the scatter of the particle positions in the swarm decreased but without further reduction of rmsPE. The final constant triplet was C/H/R = 0.2982/0.2497/0.1435. The Trust-Region-Reflective/Interior-Point algorithms showed convergence after 27/17 iterations, respectively, resulting in formula constant triplets C/H/R = 0.2982/0.2496/0.1436 and 0.2982/0.2495/0.1436, both with the same rmsPE as the PSO algorithm (rmsPE = 0.3440 D).
The PSO appears to be a powerful adaptive nonlinear iteration algorithm for formula constant optimisation even in formulae with more than 1 constant. It acts independently of an analytical or numerical gradient and is in general able to search for the best solution even with multiple local minima of the target function.
研究粒子群优化（PSO）作为一种现代的纯数据驱动的非线性迭代策略，用于优化眼内晶状体计算公式中晶状体公式常数的优化。
我们为 Castrop 公式实施了 PSO 算法，以优化数据集 N=888 例白内障眼的均方根公式预测误差（rmsPE，定义为实际屈光度与预测屈光度之差），这些眼均植入了 Hoya Vivinex 疏水性丙烯酸非球面晶状体。超参数设置为惯性：0.8，加速度 c1=c2=0.1。该算法使用 N=100 个具有随机位置和速度的粒子在常数三元组参数空间 C=0.25 到 0.45、H=-0.25 到 0.25 和 R=-0.25 到 0.25 的框约束内初始化。将该算法的性能与经典的基于梯度的信赖域反射和内点算法进行了比较。
PSO 算法在 37 次迭代后显示出快速而稳定的收敛。rmsPE 系统地降低至 0.3440 屈光度（D）。随着进一步的迭代，粒子在群中的位置的散布减小，但 rmsPE 没有进一步减小。最终的常数三元组为 C/H/R=0.2982/0.2497/0.1435。信赖域反射/内点算法分别在 27/17 次迭代后显示出收敛，得到的公式常数三元组 C/H/R=0.2982/0.2496/0.1436 和 0.2982/0.2495/0.1436，均具有与 PSO 算法相同的 rmsPE（rmsPE=0.3440 D）。
PSO 似乎是一种强大的自适应非线性迭代算法，即使在具有多个常数的公式中，也可用于公式常数优化。它独立于分析或数值梯度，并且通常能够搜索到目标函数的最佳解决方案，即使存在多个局部最小值。
