This research introduces the Deep Operator Network (DeepONet) as a robust surrogate modeling method within the context of digital twin (DT) systems for nuclear engineering. With the increasing importance of nuclear energy as a carbon-neutral solution, adopting DT technology has become crucial to enhancing operational efficiencies, safety, and predictive capabilities in nuclear engineering applications. DeepONet exhibits remarkable prediction accuracy, outperforming traditional ML methods. Through extensive benchmarking and evaluation, this study showcases the scalability and computational efficiency of DeepONet in solving a challenging particle transport problem. By taking functions as input data and constructing the operator $G$ from training data, DeepONet can handle diverse and complex scenarios effectively. However, the application of DeepONet also reveals challenges related to optimal sensor placement and model evaluation, critical aspects of real-world implementation. Addressing these challenges will further enhance the method's practicality and reliability. Overall, DeepONet presents a promising and transformative tool for nuclear engineering research and applications. Its accurate prediction and computational efficiency capabilities can revolutionize DT systems, advancing nuclear engineering research. This study marks an important step towards harnessing the power of surrogate modeling techniques in critical engineering domains.