In this paper, we propose a physics-informed machine learning method called Deep Dirac Neural Networks, based on the framework of Dirac dynamical system [10]. This method is a powerful tool for learning a generalized energy function in mechanical systems with constraints. Specifically, we focus on mechanical systems with holonomic constraints. Our approach enables the learning of not only dynamic entities such as the generalized energy but also kinematic entities such as Lagrange multipliers and holonomic constraint functions, directly from training and target data, without requiring any prior knowledge of the constraint conditions. This is achieved by training three separate modular models in the framework of the Dirac dynamical systems, where a loss function is designed to enforce the holonomic constraints. Unlike conventional approaches, our method does not rely on predefined constraint conditions, offering greater flexibility in modeling. Finally, we demonstrate the effectiveness of the proposed method through numerical experiments using a double pendulum as an illustrative example.