论文标题
使用神经网络加速高阶不连续的Galerkin求解器：3D可压缩Navier-Stokes方程
Accelerating high order discontinuous Galerkin solvers using neural networks: 3D compressible Navier-Stokes equations
论文作者
论文摘要
我们建议使用神经网络加速高阶不连续的Galerkin求解器。我们包括强迫低多项式模拟的纠正措施以提高其精度。强迫是通过使用高多项式模拟的深度完全连接的神经网络来获得的，但仅在短时间内训练。通过这种纠正效率，我们可以更快地运行低多项式订单模拟（以较大的时间步长和每次时间步的低成本），同时提高其准确性。 我们探索了这个想法，该想法是（Marique and Ferrer，CAF 2022）中的一维汉堡方程式，并将这项工作扩展到了3D Navier-Stokes方程，具有和没有大型涡流模拟封闭模型。我们用湍流的泰勒绿色涡旋情况和各种雷诺数（30、200和1600）测试方法。此外，随着时间的流逝，泰勒绿色涡流随时间而演变，并涵盖了层流，过渡和湍流方案。 事实证明，提出的方法适用于各种流量和政权。结果表明，矫正强迫在所有雷诺数字和时间范围内都是有效的（不包括初始流量开发）。我们可以用8个多项式序列训练纠正措施，以在训练时间范围之外纠正时，将模拟的精度从多项式订单3到6提高。低阶正确解决方案的速度比具有可比精度的模拟快4至5倍（多项式顺序6）。 此外，我们探索超参数的变化，并使用转移学习来加快培训。我们观察到，使用不同的流条件训练纠正效率没有用。但是，已经训练有素的纠正措施可用于初始化新的培训（在正确的流条件下），以获得有效的强迫，只有少量训练迭代。
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.
