Advancing Computational Accuracy and Efficiency through Machine Learning in Numerical Analysis

The power from machine learning based solution strategies is shown by using them for hybridizing computational models with data-driven exploratory features. The findings provide theoretical underpinnings and algorithms for practical computational methods to support better simulations, addressing important mathematical problems arising in diverse areas. Areas for future research can be hybrid algorithms and the extension of these techniques to multiscale/Multiphysics problems. Theoretical contributions to numerical analysis and approximation theory lead to more precise computational solutions for difficult problems. The methodology of the study extends to new methods and algorithms for better tenths in numerical approximations such as non-polynomial basis functions, adaptive approximation schemes, or machine learning-based techniques. These methods provide improved accuracy and computational efficiency in function approximation, quadrature rules, the solution of differential equations compared to traditional approaches. The results indicate the effectiveness of these methods for improving accuracy and efficiency in many fields. The paper contains a thorough discussion of challenges from the numerical analysis and approximation theory viewpoint, with special emphasis on computational accuracy vs. efficiency trade-off maneuverability. New approaches and algorithms are also studied in the context of numerical approximation that focus on improving precision (e.g. function approximation, quadrature rules) or efficiency (solutions to differential equations). This includes using machine learning methods for fast, accurate and adaptive approximation schemes, which may include non-polynomial basis functions. We also analyze the errors and compute the computational complexities of our results, and give some concrete applications to problems in physics, engineering and finance. In all cases, they are conducted within the frame of ethical considerations and data management and then presented in three key sections: function approximation, quadrature rules & numerical integration as well as the solution of differential equations.