Machine learning methods have been successful in many areas, like image classification and natural language processing. However, it still needs to be determined how to apply ML to areas with mathematical constraints, like solving PDEs. Among various approaches to applying ML techniques to solving PDEs, the data-driven discretization method presents a promising way of accelerating and improving existing PDE solver on structured grids where it predicts the coefficients of quasi-linear stencils for computing values or derivatives of a function at given positions. It can improve the accuracy and stability of low-resolution simulation compared with using traditional finite difference or finite volume schemes. Meanwhile, it can also benefit from traditional numerical schemes like achieving conservation law by adapting finite volume type formulations. In this thesis, we have implemented the shallow water equation and Euler equation classic solver under a different framework. Experiments show that our classic solver performs much better than the Pyclaw solver. Then we propose four different deep neural networks for the ML-based solver. The results indicate that two of these approaches could output satisfactory solutions.

翻译：机器学习方法已在诸多领域取得成功，例如图像分类与自然语言处理。然而，如何将机器学习应用于具有数学约束的领域（如求解偏微分方程）仍有待探索。在将机器学习技术应用于求解偏微分方程的各种方法中，数据驱动的离散化方法提供了一种极具前景的途径，可在结构化网格上加速并改进现有偏微分方程求解器。该方法通过预测拟线性模板的系数，以计算给定位置处的函数值或导数值。与传统有限差分或有限体积格式相比，该方法能提升低分辨率模拟的精度与稳定性。同时，其亦可受益于传统数值格式，例如通过采用有限体积型公式实现守恒律。本论文在不同框架下实现了浅水方程与欧拉方程的经典求解器。实验表明，我们的经典求解器性能显著优于Pyclaw求解器。随后，我们为基于机器学习的求解器提出了四种不同的深度神经网络架构。结果表明，其中两种方法能够输出令人满意的解。