单元学习：一种通过机器学习加速有限元类方法并应用于辐射传输的系统性方法
Element learning: a systematic approach of accelerating finite element-type methods via machine learning, with applications to radiative transfer
AI总结：
本研究提出单元学习方法，通过神经网络逐单元学习偏微分方程解映射，替代HDG的局部求解器，无需重训练即可适配不同几何与参数，在辐射传输方程测试中实现5到10倍加速。
摘要：
本文提出了一种通过机器学习加速有限元类方法的系统性方法，用于偏微分方程（PDE）的数值求解。其核心思想是使用神经网络以逐单元的方式学习偏微分方程的解映射。该映射以单元几何形状和该单元上的偏微分方程参数作为输入，输出两个算子：（1）用于单元间通信的in2out算子，（2）用于逐单元解恢复的in2sol算子（格林函数）。该方法的一个显著优势是，一旦训练完成，该网络无需重新训练即可用于任意区域几何形状和任意参数分布下的偏微分方程数值求解。此外，由于训练是在单元层面而非整个区域上进行的，训练过程也显著简化。我们将这种方法称为单元学习（element learning）。该方法与可杂交间断伽辽金（HDG）方法密切相关，其本质是用机器学习方法替代了HDG的局部求解器。我们针对一个示例偏微分方程即辐射传输方程开展了数值测试，测试涵盖多种场景，包括理想化或真实云场，以及云边界过渡处具有平滑或陡峭梯度的情况。在相对L²误差为10⁻³的固定精度水平下，且每个单元的多项式次数p=6时，我们观察到与经典有限元类方法相比，单元学习方法的加速比约为5到10倍。
英文摘要：
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.
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