In this paper, we propose a new deep learning method, named finite volume method (DFVM) to solve high-dimension partial differential equations (PDEs). The key idea of DFVM is that we construct a new loss function under the framework of the finite volume method. The weak formulation makes DFVM more feasible to solve general high dimensional PDEs defined on arbitrarily shaped domains. Numerical solutions obtained by DFVM also enjoy physical conservation property in the control volume of each sampling point, which is not available in other existing deep learning methods. Numerical results illustrate that DFVM not only reduces the computation cost but also obtains more accurate approximate solutions. Specifically, for high-dimensional linear and nonlinear elliptic PDEs, DFVM provides better approximations than DGM and WAN, by one order of magnitude. The relative error obtained by DFVM is slightly smaller than that obtained by PINN, but the computation cost of DFVM is an order of magnitude less than that of the PINN. For the time-dependent Black-Scholes equation, DFVM gives better approximations than PINN, by one order of magnitude.

翻译：本文提出了一种新的深度学习方法，即有限体积法（DFVM），用于求解高维偏微分方程（PDE）。DFVM的核心思想是在有限体积法框架下构造新的损失函数。弱形式使DFVM更适用于求解定义在任意形状域上的通用高维PDE。DFVM获得的数值解在每个采样点的控制体积内具有物理守恒特性，这是现有其他深度学习方法所不具备的。数值结果表明，DFVM不仅降低了计算成本，而且能获得更精确的近似解。具体而言，对于高维线性和非线性椭圆型PDE，DFVM的近似精度比DGM和WAN高一个数量级。DFVM的相对误差略小于PINN，但其计算成本比PINN低一个数量级。对于时间依赖的Black-Scholes方程，DFVM的近似精度比PINN高一个数量级。