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. In this work, we develop a model-constrained discontinuous Galerkin Network (DGNet) approach, a significant extension to our previous work [Model-constrained Tagent Slope Learning Approach for Dynamical Systems], 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; (ii) employing a model-constrained approach to ensure the learned tangent slope satisfies governing equations; (iii) utilizing a GNN-inspired architecture 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; (iv) implementing the input normalization technique that allows surrogate models to generalize across different initial conditions, geometries, meshes, boundary conditions, and solution orders; 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 effectiveness, stability, and generalizability of our novel DGNet approach, we present comprehensive numerical results for 1D and 2D compressible Euler equation problems.

翻译：在实际工程与科学应用（尤其是数字孪生场景）中，控制、优化、不确定性量化和决策制定亟需大规模复杂动力系统的实时精确解。本文针对可压缩欧拉方程，提出一种模型约束间断伽辽金网络（DGNet）方法，这是我们前期工作[面向动力系统的模型约束切线斜率学习方法]的重要拓展，旨在实现分布外泛化。DGNet 的核心在于多项关键策略的协同：（i）利用时间积分方案捕捉时间相关性，并借助神经网络速度优势降低计算耗时；（ii）采用模型约束方法确保学习到的切线斜率满足控制方程；（iii）构建受图神经网络启发的架构，其中边表示黎曼求解器代理模型，节点表示体积积分修正代理模型，从而具备捕捉间断能力、降低混叠误差并实现网格离散化泛化；（iv）实施输入归一化技术，使代理模型能够泛化至不同初始条件、几何构型、网格、边界条件和解的阶次；（v）引入数据随机化技术，该技术不仅隐式地促进代理模型与真实数值模型直至二阶导数的一致性，从而确保长期稳定性和预测能力，还在训练过程中充当数据生成引擎，提升对未见数据的泛化性能。为验证我们提出的 DGNet 方法的有效性、稳定性和泛化能力，本文针对一维和二维可压缩欧拉方程问题给出了全面的数值结果。