enhanced milne-simpson s methods for autonomous and singular differential equations

Solving autonomous and singular differential equations remains a persistent challenge for traditional numerical methods due to the presence of critical points and singularities that degrade solution accuracy. this paper introduces a novel hybrid framework that uniquely integrates the classical milne-simpson’s method with a neural network-based refinement strategy to address these challenges. while milne-simpson’s method provides an efficient initial approximation, its accuracy deteriorates near singular behaviors. to overcome this, we propose a deep learning-based post-processing stage specifically designed to refine the coarse numerical solutions. unlike previous works that either apply neural networks as standalone solvers or generic correctors, our approach explicitly tailors the neural architecture to learn correction functions that complement the structural dynamics of milne-simpson’s output. the neural network is trained on synthetic datasets generated to highlight the failure modes of classical methods, particularly focusing on complex autonomous and singular behavior. experimental evaluations demonstrate that our hybrid approach significantly improves solution accuracy in problematic regions without compromising computational efficiency, thus offering a robust and scalable method for solving challenging differential equations.