From Mathematical Modeling and Simulation to Digital Twins: Bridging Theory and Digital Realities in Industry and Emerging Technologies

Against the background of the unprecedented advancements related to Industry 4.0 and beyond, transitioning from classical mathematical models to fully embodied digital twins represents a critical change in the planning, monitoring, and optimization of complex industrial systems. This work outlines the subject within the broader field of applied mathematics and computational simulation while highlighting the critical role of sound mathematical foundations, numerical methodologies, and advanced computational tools in creating data-informed virtual models of physical infrastructures and processes in real time. The discussion includes examples related to smart manufacturing, additive manufacturing technologies, and cyber–physical systems with a focus on the potential for collaboration between physics-informed simulations, data unification, and hybrid machine learning approaches. Central issues including a lack of scalability, measuring uncertainties, inter-operability challenges, and ethical concerns are discussed along with relevant considerations for the Digital Twin.