Machine Learning-Assisted Numerical Integration for Predictive Mathematical Modelling

Abstract

Accurate numerical integration remains a fundamental challenge in predictive mathematical modelling, particularly for nonlinear, high-dimensional, and data-driven systems. Conventional numerical integration techniques, including the Trapezoidal Rule, Simpson's Rule, and Gaussian Quadrature, are widely used to approximate integrals that cannot be solved analytically. However, their performance often depends on integration parameters, data quality, and computational resources, particularly for nonlinear, high-dimensional, and uncertain systems. This study proposes a hybrid system that integrates machine learning with numerical integration to enhance prediction accuracy, computational efficiency, and numerical robustness. The designed system uses numerical integration to model continuous system dynamics while machine learning algorithms learn from historical and simulated datasets to select suitable integration methods, optimize step sizes, and estimate numerical errors. Performance was evaluated using MAE, RMSE, MAPE, convergence rate, computational time, stability analysis, robustness analysis, and statistical significance testing. It reduced approximation errors by over 80%, improved predictive accuracy to approximately 99%, and maintained numerical stability across benchmark problems. The results demonstrate that machine learning-assisted numerical integration significantly improves computational efficiency and predictive performance while preserving the convergence and stability properties of classical numerical methods. The study develops a novel adaptive machine learning-assisted numerical integration algorithm that automatically selects numerical integration strategies while preserving theoretical convergence and stability across science, engineering, finance, and environmental systems.