In this paper, we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations (PDEs). The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion. This map takes input of the element geometry and the PDEs' parameters on that element, and gives output of two operators -- (1) the in2out operator for inter-element communication, and (2) the in2sol operator (Green's function) for element-wise solution recovery. A significant advantage of this approach is that, once trained, this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining. Also, the training is significantly simpler since it is done on the element level instead on the entire domain. We call this approach element learning. This method is closely related to hybridizbale discontinuous Galerkin (HDG) methods in the sense that the local solvers of HDG are replaced by machine learning approaches. Numerical tests are presented for an example PDE, the radiative transfer equation, in a variety of scenarios with idealized or realistic cloud fields, with smooth or sharp gradient in the cloud boundary transition. Under a fixed accuracy level of $10^{-3}$ in the relative $L^2$ error, and polynomial degree $p=6$ in each element, we observe an approximately 5 to 10 times speed-up by element learning compared to a classical finite element-type method.

翻译：本文提出一种系统方法，通过机器学习加速有限元类方法求解偏微分方程(PDEs)的数值解。其核心思想是采用神经网络以逐元素方式学习PDE的解映射。该映射以单元几何形状及该单元上的PDE参数为输入，输出两个算子——(1)用于单元间通信的in2out算子，以及(2)用于单元级解恢复的in2sol算子（格林函数）。该方法的一个显著优势是：一旦训练完成，该网络可在无需重新训练的条件下，适用于任意域几何形状和任意参数分布的PDE数值求解。此外，由于训练在单元层级而非全域进行，训练过程显著简化。我们将此方法称为元素学习。该方法与混合间断伽辽金（HDG）方法密切相关，本质上是用机器学习方法替代了HDG中的局部求解器。针对辐射传输方程这一示例PDE，我们在多种场景（包括理想化或真实云场、云边界过渡区光滑或陡峭梯度）下进行了数值测试。在相对L²误差精度固定为$10^{-3}$、各单元多项式阶数$p=6$的条件下，相比经典有限元类方法，元素学习实现了约5至10倍的加速比。