Problem optimization of ray tracing through the crystalline lens of the eye with an artificial neural network and Grey Wolf optimizer

Abstract

Ray tracing through the crystalline lens of the eye is a complex optical problem traditionally tackled by mathematical techniques that may lack optimal accuracy. This paper introduces a novel method that integrates the Grey Wolf optimizer (GWO) with an artificial neural network (ANN), termed ANNGWO, to enhance the precision of ray tracing through the lens. The ANNGWO approach involves defining a cost function with a multi-layer neural network incorporating three activation functions: Log-sigmoid, Radial basis, and Tan-sigmoid. GWO is then employed to optimize this cost function, effectively simulating the intricate geometry of the crystalline lens. The methodology includes designing a multi-layer ANN with the activation function to model the ray tracing problem.This is followed by constructing the cost function by formulating the unsupervised error function for the equation and its initial conditions. Subsequently, GWO is utilized to minimize this cost function, thereby enhancing the accuracy of the simulation. The performance of ANNGWO is compared with traditional methods, including the Laplace decomposition method (LDM), multi-step differential transform method (MDTM), and the fourth-order Runge–Kutta method (RKM). Numerical experiments demonstrate that ANNGWO significantly improves accuracy. For instance, the mean absolute error (AE) using the Log-sigmoid activation function is $6.869\times 10^{-2}$ for 10 neurons. Best-case AE values range from $3.390\times 10^{-4}$ to $1.169\times 10^{-2}$, outperforming traditional methods. Additionally, ANNGWO exhibits consistently lower standard deviation of absolute error across all time steps, reflecting enhanced stability. Compared to LDM, MDTM, and RKM, ANNGWO achieves a significant reduction in error, underscoring its superior precision and reliability. The method’s ability to avoid local minima and its robust convergence behavior make it a powerful tool for simulating light movement in the eye, with promising applications in vision research and ocular imaging.