This paper presents a novel physics-informed recurrent neural network (PI-RNN) modeling methodology for district heating systems (DHSs), enabling accurate and computationally efficient dynamic models. The proposed approach integrates operational data with physical knowledge of the DHS network topology to enhance model accuracy and training efficiency. The PI-RNN architecture is designed to mimic the physical structure of the DHS, with multiple RNNs interconnected according to the network's topology. This methodology allows for faster training and improved modeling performance compared to standard RNNs, even when using reduced-order models. The developed PI-RNN is then used to design a nonlinear model predictive control (NMPC) strategy, which optimizes DHS operation by minimizing production costs, increasing system efficiency, and respecting operational constraints. The proposed methods are tested on a DHS benchmark, showing promising results in both modeling and control. The work addresses the challenges of modeling and controlling large-scale DHS systems, which are governed by complex nonlinear equations and require significant computational resources. By leveraging physical insights and operational data, the PI-RNN approach provides a computationally efficient and accurate alternative to traditional physical models, enabling effective NMPC for optimal DHS operation.This paper presents a novel physics-informed recurrent neural network (PI-RNN) modeling methodology for district heating systems (DHSs), enabling accurate and computationally efficient dynamic models. The proposed approach integrates operational data with physical knowledge of the DHS network topology to enhance model accuracy and training efficiency. The PI-RNN architecture is designed to mimic the physical structure of the DHS, with multiple RNNs interconnected according to the network's topology. This methodology allows for faster training and improved modeling performance compared to standard RNNs, even when using reduced-order models. The developed PI-RNN is then used to design a nonlinear model predictive control (NMPC) strategy, which optimizes DHS operation by minimizing production costs, increasing system efficiency, and respecting operational constraints. The proposed methods are tested on a DHS benchmark, showing promising results in both modeling and control. The work addresses the challenges of modeling and controlling large-scale DHS systems, which are governed by complex nonlinear equations and require significant computational resources. By leveraging physical insights and operational data, the PI-RNN approach provides a computationally efficient and accurate alternative to traditional physical models, enabling effective NMPC for optimal DHS operation.