Combining Evolutionary Optimization and Deep Learning for Inverse Mathematical Problems

Abstract

This study proposes a hybrid Evolutionary Optimization–Deep Learning (EO-DL) framework to improve solution accuracy, robustness, and convergence. The framework combines deep neural networks with three evolutionary optimization algorithms—Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Differential Evolution (DE). Deep learning models inverse mappings, while evolutionary algorithms optimize network parameters. The approach is evaluated on benchmark inverse problems under varying noise levels using reconstruction accuracy, Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), convergence, and computational efficiency. The EO-DL framework outperformed conventional optimization, deep learning, and physics-informed neural network methods. Differential Evolution achieved the best performance, with 97. 3% reconstruction accuracy, the lowest error metrics, faster convergence, and greater robustness to noise. The proposed EO-DL framework offers a scalable and reliable solution for complex inverse mathematical problems with applications in scientific computing, engineering, medical imaging, and data-driven modeling.