This paper develops an advanced framework for the operational optimization of integrated multi-energy systems that encompass electricity, gas, and heating networks. Introducing a cutting-edge stochastic gradient-enhanced distributionally robust optimization approach, this study integrates deep learning models, especially generative adversarial networks, to adeptly handle the inherent variability and uncertainties of re-newable energy and fluctuating consumer demands. The effectiveness of this framework is rigorously tested through detailed simulations mirroring real-world urban energy consumption, renewable energy production, and market price fluctuations over an annual period. The results reveal substantial improvements in the resilience and efficiency of the grid, achieving a reduction in power distribution losses by 15% and enhancing voltage stability by 20%, markedly outperforming conventional systems. Additionally, the framework facilitates up to 25% in cost reductions during peak demand periods, significantly lowering operational costs. The adoption of stochastic gradients further refines the framework's ability to continually adjust to real-time changes in environmental and market conditions, ensuring stable grid operations and fostering active consumer engagement in demand-side management. This strategy not only aligns with contemporary sustainable energy practices but also provides scalable and robust solutions to pressing challenges in modern power network management.