论文标题
通过机器学习增强冲击方法
Enhancement of shock-capturing methods via machine learning
论文作者
论文摘要
近年来，机器学习已被用来为算法解决方案以及对现有算法进行微调的算法解决方案的问题创建数据驱动的解决方案。这项研究将机器学习应用于开发改进的有限体积方法，用于使用不连续的解决方案模拟PDE。冲击捕获方法利用了不能保证最佳的非线性切换功能。由于数据可用于学习非线性关系，因此我们训练神经网络以改善五阶Weno方法的结果。我们后处理神经网络的输出，以确保该方法是一致的。训练数据包括一组可集成函数的单元平均值和插值值之间的精确映射，这些函数代表了我们在模拟PDE时期望看到的波形。我们演示了我们关于不连续函数的线性对流的方法，Inviscid汉堡方程和1-D Euler方程。对于后者，我们检查了用于湍流震荡相互作用的Shu-Osher模型问题。我们发现，我们的方法在模拟中优于WENO，在模拟中，由于数值粘度，数值解的过多扩散。
In recent years, machine learning has been used to create data-driven solutions to problems for which an algorithmic solution is intractable, as well as fine-tuning existing algorithms. This research applies machine learning to the development of an improved finite-volume method for simulating PDEs with discontinuous solutions. Shock capturing methods make use of nonlinear switching functions that are not guaranteed to be optimal. Because data can be used to learn nonlinear relationships, we train a neural network to improve the results of a fifth-order WENO method. We post-process the outputs of the neural network to guarantee that the method is consistent. The training data consists of the exact mapping between cell averages and interpolated values for a set of integrable functions that represent waveforms we would expect to see while simulating a PDE. We demonstrate our method on linear advection of a discontinuous function, the inviscid Burgers' equation, and the 1-D Euler equations. For the latter, we examine the Shu-Osher model problem for turbulence-shockwave interactions. We find that our method outperforms WENO in simulations where the numerical solution becomes overly diffused due to numerical viscosity.
