The use of digital twin and shadow concepts for industrial material processes has introduced new approaches to bridge the gap between physical and cyber manufacturing processes. Consequently, many multidisciplinary areas, such as advanced sensor technologies, material science, data analytics, and machine learning algorithms, are employed to create these hybrid systems. Meanwhile, new additive manufacturing (AM) processes for metals and polymers, based on emerging technologies, have shown promise for the manufacturing of sophisticated parts with complex geometries. These processes are undergoing a major transformation with the advent of digital technology, hybrid physical-data-driven modeling, and fast-reduced models. This study presents a fresh perspective on hybrid physical-data-driven and reduced order modeling (ROM) techniques for the digitalization of AM processes within a digital twin concept. The main contribution of this study is to demonstrate the benefits of ROM and machine learning (ML) technologies for process data handling, optimization/control, and their integration into the real-time assessment of AM processes. Therefore, a novel combination of efficient data-solver technology and an architecturally designed neural network (NN) module is developed for transient manufacturing processes with high heating/cooling rates. Furthermore, a real-world case study is presented, showcasing the use of hybrid modeling with ROM and ML schemes for an industrial wire arc AM (WAAM) process.