A Comprehensive Study on Predicting Numerical Integration Errors using Machine Learning Approaches

Abstract

Numerical integration methods, such as Trapezoidal, Simpson’s, and Gaussian quadrature, are fundamental in approximating definite integrals when closed-form solutions are intractable. However, these approximations inherently involve error terms influenced by factors such as function behavior, discretization granularity, and method order. Classical error estimation techniques rely on analytical bounds, which may not always capture complex or irregular function behaviors. In this study, we propose a novel integration of machine learning (ML) models particularly regression-based and tree-based algorithms—to predict numerical integration errors with improved accuracy and generalization. By training ML models on curated datasets that include function features, step sizes, and actual errors derived from classical integration methods, we establish a predictive framework that learns error patterns from empirical data rather than relying solely on theoretical bounds. Our methodology is validated using benchmark integrable functions from standard mathematical libraries. We demonstrate that ML approaches, especially Gradient Boosting Regression and Support Vector Machines, outperform traditional heuristic error bounds in terms of Root Mean Square Error (RMSE) and coefficient of determination (R²). This hybridization of numerical analysis and data-driven learning opens pathways for adaptive integration schemes and intelligent numerical computing. Our research signifies an important interdisciplinary development that enhances the reliability of numerical integration in computational mathematics, physics simulations, and engineering systems.