Control optimization is essential to achieve high performance and cost efficiency in large-scale physical systems such as inertial district energy networks. These systems play a key role in mitigating climate change, particularly in heating and cooling. They integrate multiple energy sources and require intelligent control strategies to minimize costs while preserving high efficiency. However, the complexity of their underlying dynamics and the high computational load associated with their numerical simulation often make predictive control prohibitively slow or limited to short time horizons. In this work, we introduce a hybrid modeling framework where predictive control is accelerated using a physics-informed spatio-temporal graph neural network as a state-space surrogate model. Unlike existing models, our approach incorporates first-principle conservation laws to improve accuracy and generalization. This approach drastically reduces simulation time by four orders of magnitude, enabling faster decision-making. Using real-world data, we introduce a time-series augmentation technique combining Gaussian scaling and time slicing to improve model performance. Extensive experiments were conducted to evaluate the accuracy and generalization capacity of the learned model. Once validated, several optimization techniques were implemented, including evolutionary algorithms and reinforcement learning, which are assessed against rule-based control. Results show that this approach enables scalable predictions and efficient control, achieving up to 29% energy cost savings during mid-season while cutting optimization time from days to mere minutes.