Deep learning and Partial Differential Equations

在计算机科学和工程领域，从固体力学和流体力学等关键领域到视觉效果、计算机图形学、动画和虚拟现实等娱乐/教育领域，有限元方法在求解偏微分方程组中得到了广泛的应用。这类研究的一个关键的基本方面是在精度和速度之间取得良好的平衡：例如，很大一部分工作一直致力于生成高质量和自适应的有限元网格。在机器学习和深度学习的时代，科学家们一直在积极探索数据驱动的方法，如深度神经网络，以帮助直接求解偏微分方程组。一个可能的方向是通过深度学习将传统的数值求解器与智能生成的有限元网格相结合。这样，我们的目标是既加快速度，又保证准确性。该方法消除了深度学习模型直接用于预测解时，由于训练数据有限或缺乏模型泛化能力而产生错误预测的最坏情况。本项目以此为出发点，旨在探索深度学习和有限元方法之间的相互作用，以加速求解偏微分方程组。其目的是开发新的方法来加速求解偏微分方程组的不同阶段，例如自适应网格生成、高质量的非线性求解器的初始迭代、最优预条件等。由于该项目将探索将深度学习与偏微分方程组相结合的新方法，因此在上述领域具有潜在的应用前景。

Finite Element Methods (FEM) have been ubiquitously used in solving Partial Differential Equations (PDEs in an extremely wide range of fields in computer science and engineering, ranging from critical domains such as solid mechanics and fluid dynamics, to entertainments/education such as visual effects, computer graphics, animation and virtual reality. One key fundamental aspect to such research is a well-balanced trade-off between accuracy and speed: for example, a significant proportion of the effort has been devoted to generating high-quality and adaptive FEM meshes.In the era of machine learning and deep learning, scientists have been actively exploring data-driven methods such as deep neural networks to help directly solve PDEs. One possible direction is to combine traditional numerical solvers with smartly generated FEM meshes via deep learning. This way, we aim to both accelerate the speed and safeguard the accuracy. This route eliminates the worst-case scenarios where deep learning models can generate wrong predictions due to limited training data or the lack of model generalizability when used to predict the solutions directly.Taking the aforementioned as a starting point, the project aims to explore the interplay between deep learning and FEM methods to accelerate solving PDEs. The objective is to develop new methods that can speed up different stages of solving PDEs, e.g. adaptive mesh generation, high quality initial iterates for nonlinear solvers, optimal preconditioners, etc. Since the project will investigate new ways of combining deep learning and PDEs, there are potential applications in a wide range of fields such as the mentioned above.