使用神经网络加速高阶间断伽辽金求解器：三维可压缩Navier-Stokes方程
Accelerating high order discontinuous Galerkin solvers using neural networks: 3D compressible Navier-Stokes equations
AI总结：
提出使用神经网络训练校正强迫项来加速低阶间断伽辽金求解器，在三维Navier-Stokes方程中验证了该方法能有效提升多雷诺数下的模拟精度，且比同等精度的高阶模拟快4-5倍。
摘要：
我们提出使用神经网络来加速高阶间断伽辽金（discontinuous Galerkin）求解器。我们向低多项式阶模拟中加入校正强迫项以提升其精度。该强迫项通过训练深度全连接神经网络获得，训练时使用高多项式阶模拟，但仅用于较短的时间范围。借助该校正强迫项，我们能够更快地运行低多项式阶模拟（具有更大的时间步长和更低的时间步成本），同时提高其精度。\n我们在(Marique and Ferrer, CAF 2022)中针对一维Burgers方程探索了这一想法，并已将此项工作扩展至三维Navier-Stokes方程，包含和不包含大涡模拟（Large Eddy Simulation）闭合模型的情况。我们使用湍流Taylor Green Vortex算例，并针对多种雷诺数（30、200和1600）对该方法进行了测试。此外，Taylor Green Vortex随时间演化，随着时间推移覆盖了层流、过渡流和湍流状态。\n所提出的方法被证明适用于多种流动和状态。结果表明，校正强迫项在所有雷诺数和时间范围内（不包括初始流动发展阶段）均有效。我们可以使用多项式阶数为8来训练校正强迫项，以在训练时间范围之外进行校正时，将多项式阶数从3提高到6的模拟精度提升。低阶校正解比具有相当精度（多项式阶数6）的模拟快4到5倍。\n此外，我们探索了超参数的变化，并使用迁移学习来加速训练。我们观察到，使用不同的流动条件来训练校正强迫项是没有用的。然而，已经训练好的校正强迫项可以用来初始化新的训练（在正确的流动条件下），以仅通过几次训练迭代获得有效的强迫项。
英文摘要：
We propose to accelerate a high order discontinuous Galerkin solver using neural networks. We include a corrective forcing to a low polynomial order simulation to enhance its accuracy. The forcing is obtained by training a deep fully connected neural network, using a high polynomial order simulation but only for a short time frame. With this corrective forcing, we can run the low polynomial order simulation faster (with large time steps and low cost per time step) while improving its accuracy. We explored this idea for a 1D Burgers' equation in (Marique and Ferrer, CAF 2022), and we have extended this work to the 3D Navier-Stokes equations, with and without a Large Eddy Simulation closure model. We test the methodology with the turbulent Taylor Green Vortex case and for various Reynolds numbers (30, 200 and 1600). In addition, the Taylor Green Vortex evolves with time and covers laminar, transitional, and turbulent regimes, as time progresses. The proposed methodology proves to be applicable to a variety of flows and regimes. The results show that the corrective forcing is effective in all Reynolds numbers and time frames (excluding the initial flow development). We can train the corrective forcing with a polynomial order of 8, to increase the accuracy of simulations from a polynomial order 3 to 6, when correcting outside the training time frame. The low order correct solution is 4 to 5 times faster than a simulation with comparable accuracy (polynomial order 6). Additionally, we explore changes in the hyperparameters and use transfer learning to speed up the training. We observe that it is not useful to train a corrective forcing using a different flow condition. However, an already trained corrective forcing can be used to initialise a new training (at the correct flow conditions) to obtain an effective forcing with only a few training iterations.
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