COMPUTATIONAL NUMERICAL INTEGRATION OF REAL AND COMPLEX MULTIVARIABLE FUNCTIONS: AI INTEGRATED METHODS AND APPLICATIONS IN MODERN ENGINEERING

Abstract

Computational numerical integration is fundamental to real and complex multivariable problems that define the modern artificial intelligence (AI) based decision models. With the increasing data intensity, nonlinearity, and dimensionality of engineering systems, the efficiency of numerical integration methods is of paramount importance for enhancing the numerical accuracy, optimization performance, uncertainty quantification, and predictable reliability of engineering systems. The mathematical principles of numerical integration are reviewed, and deterministic, stochastic,  adaptive and hybrid integration techniques in the context of AI-based computational frameworks are explored. A particular focus is on applications in engineering, such as computational fluid dynamics, energy management, digital twin, control engineering, and large-scale engineering simulations, among others. Combining numerical methods with machine learning, probabilistic modelling, and intelligent optimization allows efficient analysis of complex engineering systems and robust and data-driven decision making. The scalability and efficiency of engineering decision models is discussed in relation to recent advancements of high-performance computing, GPU acceleration, parallel processing, digital twins, and explainable AI. High computational complexity, numerical instability, high-dimensional processing, scalability and model interpretability are thoroughly discussed as key challenges. Lastly, new trends that feature AI-driven adaptive integration, intelligent optimization, real-time simulation, and self-governing computational systems are emphasized. In conclusion, the combination of advanced numerical techniques and AI offers a solid stepping stone towards the creation of accurate, scalable, and intelligent decision-support systems that can tackle increasingly complex engineering challenges and facilitate next-generation industrial automation and technological innovations.