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 consist 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–shock wave interactions. We find that our method outperforms WENO in simulations where the numerical solution becomes overly diffused due to numerical viscosity.

中文翻译：

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通过机器学习增强冲击捕捉方法

近年来，机器学习已被用于为算法解决方案难以解决的问题创建数据驱动的解决方案，以及对现有算法进行微调。本研究将机器学习应用于开发改进的有限体积方法，用于模拟具有不连续解的偏微分方程。冲击捕获方法利用不能保证最佳的非线性切换函数。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进五阶 WENO 方法的结果。我们对神经网络的输出进行后处理以保证该方法是一致的。训练数据包含一组可积函数的单元平均值和内插值之间的精确映射，这些函数表示我们在模拟 PDE 时期望看到的波形。我们演示了关于不连续函数的线性平流、无粘伯格斯方程和一维欧拉方程的方法。对于后者，我们研究了湍流-冲击波相互作用的 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，由于数值粘性，数值解变得过度扩散。