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Real-time accurate solutions of large-scale complex dynamical systems are critically needed for control, optimization, uncertainty quantification, and decision-making in practical engineering and science applications, particularly in digital twin contexts. Recent research on hybrid approaches combining numerical methods and machine learning in end-to-end training has shown significant improvements over either approach alone. However, using neural networks as surrogate models generally exhibits limitations in generalizability over different settings and in capturing the evolution of solution discontinuities. In this work, we develop a model-constrained discontinuous Galerkin Network DGNet approach, a significant extension to our previous work, for compressible Euler equations with out-of-distribution generalization. The core of DGNet is the synergy of several key strategies: (i) leveraging time integration schemes to capture temporal correlation and taking advantage of neural network speed for computation time reduction. This is the key to the temporal discretization-invariant property of DGNet; (ii) employing a model-constrained approach to ensure the learned tangent slope satisfies governing equations; (iii) utilizing a DG-inspired architecture for GNN where edges represent Riemann solver surrogate models and nodes represent volume integration correction surrogate models, enabling capturing discontinuity capability, aliasing error reduction, and mesh discretization generalizability. Such a design allows DGNet to learn the DG spatial discretization accurately; (iv) developing an input normalization strategy that allows surrogate models to generalize across different initial conditions, geometries, meshes, boundary conditions, and solution orders. In fact, the normalization is the key to spatial discretization-invariance for DGNet; and (v) incorporating a data randomization technique that not only implicitly promotes agreement between surrogate models and true numerical models up to second-order derivatives, ensuring long-term stability and prediction capacity, but also serves as a data generation engine during training, leading to enhanced generalization on unseen data. To validate the theoretical results, effectiveness, stability, and generalizability of our novel DGNet approach, we present comprehensive numerical results for 1D and 2D compressible Euler equation problems, including Sod Shock Tube, Lax Shock Tube, Isentropic Vortex, Forward Facing Step, Scramjet, Airfoil, Euler Benchmarks, Double Mach Reflection, and a Hypersonic Sphere Cone benchmark.