ADVANCING COMPUTATIONAL MATHEMATICAL MODELING: INTEGRATING QUANTUM COMPUTING AND AI-DRIVEN OPTIMIZATION FOR COMPLEX SYSTEM SIMULATIONS

Abstract

Over the years, computational mathematical modeling has become an important tool in many fields of science, making use of contemporary numerical methods,AI and optimization techniques. This research investigates the underlying computational methodologies in practical mathematical modeling, spanning finite element techniques, differential equation solvers, as well as machine learning-based prediction models. Theoretical content covered includes probability theory, statistical modeling and differential equations, all used to represent
complex real-world systems. The Diedrichs study exposes the use of computational models in various disciplines such as physics, engineering, biology, medicine,finance, and environmental science used for structural analysis, disease modeling, risk assessment, and climate simulation. Despite tremendous progress, challenges such as computational inefficiency, model accuracy bounds, and scalability persist. AI, quantum computing, and hybrid models are new technologies that can be potential responses to these limitations, providing opportunities for
computational efficiency and predictive accuracy improvements. Future studies should be directed towards improving computation algorithms through adaptive algorithms, combining multiple disciplines (i.e., biology, engineering) to address complex problems, and employing high-performance computing techniques. Adaptive technology's mathematical modeling is also going to evolve, thus, adding to the jumps in scientific progress (more realistic simulation or data-based decision on complex systems). With these ideas and the novel computational
paradigms that they may enable, this will guarantee the continued evolution of this field, which holds the keys to unleashing an innovation engine for science, engineering, and applied mathematics.