In this paper, we propose the deep finite volume method (DFVM), a novel deep learning method for solving %high-order (order $\geq 2$) partial differential equations (PDEs). The key idea is to design a new loss function based on the local conservation property over the so-called {\it control volumes}, derived from the original PDE. Since the DFVM is designed according to a {\it weak instead of strong} form of the PDE, it may achieve better accuracy than the strong-form-based deep learning method such as the well-known PINN, when the to-be-solved PDE has an insufficiently smooth solution. Moreover, since the calculation of second-order derivatives of neural networks has been transformed to that of first-order derivatives which can be implemented directly by the Automatic Differentiation mechanism(AD), the DFVM usually has a computational cost much lower than that of the methods which need to compute second-order derivatives by the AD. Our numerical experiments show that compared to some deep learning methods in the literature such as the PINN, DRM, and WAN, the DFVM obtains the same or higher accurate approximate solutions by consuming significantly lower computational cost. Moreover, for some PDE with a nonsmooth solution, the relative error of approximate solutions by DFVM is two orders of magnitude less than that by the PINN.

翻译：本文提出深度有限体积法（DFVM），这是一种用于求解高阶（阶数≥2）偏微分方程的新型深度学习方法。其核心思想是基于原始偏微分方程在所谓“控制体积”上的局部守恒性质，设计一种新的损失函数。由于DFVM基于偏微分方程的弱形式而非强形式进行设计，当待求解方程的解不够光滑时，其精度可能优于基于强形式的深度学习方法（如著名的PINN）。此外，由于神经网络二阶导数的计算被转化为一阶导数的计算，而一阶导数可通过自动微分机制直接实现，DFVM的计算成本通常远低于需要通过自动微分计算二阶导数的相关方法。数值实验表明，与现有文献中的PINN、DRM和WAN等深度学习方法相比，DFVM能以显著更低的计算代价获得相同或更高精度的近似解。对于某些非光滑解的偏微分方程，DFVM近似解的相对误差比PINN低两个数量级。