Optimizing the neural network and iterated function system parameters for fractal approximation using a modified evolutionary algorithm.

Fractal interpolation has gained significant attention due to its ability to model complex, self-similar structures with high precision. However, optimizing the parameters of Iterated Function System (IFS)-based fractal interpolants remains a challenging task, particularly for Rational Fractal Cubic (RFC) splines, which offer greater flexibility in shape control. In this study, we propose an evolutionary optimization strategy to enhance the accuracy and adaptability of RFC splines by optimizing their scaling factor and shape parameters using our novel Fractal Differential Evolution (FDE) algorithm. The FDE method iteratively refines the parameter space to achieve an optimal fit to the target data, demonstrating improved convergence and computational efficiency compared to traditional approaches. To validate the effectiveness of our method, we present a detailed numerical example showcasing the impact of optimized parameters on RFC spline interpolation. Furthermore, as a practical application, we develop a predictive model by approximating the FDE-optimized RFC spline using an Artificial Neural Network (ANN). The ANN is fine-tuned through the FDE algorithm to minimize the Euclidean distance between the RFC spline and the network's predictions, ensuring high accuracy. This neural network model is subsequently used for extrapolation, enabling robust predictions beyond the observed data. Our results highlight the potential of integrating fractal-based interpolation with machine learning techniques, paving the way for applications in computational geometry, image processing, and time series forecasting.