Motivated by finite volume scheme, a cell-average based neural network method is proposed. The method is based on the integral or weak formulation of partial differential equations. A simple feed forward network is forced to learn the solution average evolution between two neighboring time steps. Offline supervised training is carried out to obtain the optimal network parameter set, which uniquely identifies one finite volume like neural network method. Once well trained, the network method is implemented as a finite volume scheme, thus is mesh dependent. Different to traditional numerical methods, our method can be relieved from the explicit scheme CFL restriction and can adapt to any time step size for solution evolution. For Heat equation, first order of convergence is observed and the errors are related to the spatial mesh size but are observed independent of the mesh size in time. The cell-average based neural network method can sharply evolve contact discontinuity with almost zero numerical diffusion introduced. Shock and rarefaction waves are well captured for nonlinear hyperbolic conservation laws.

翻译：以有限的体积计划为动力,提出一个基于细胞平均神经网络方法。该方法基于部分差异方程式的整体或薄弱配方。一个简单的前方种子网络被迫学习两个相邻时间步骤之间的解决方案平均演变过程。进行离线监督培训是为了获得最佳网络参数组,该参数组独有地识别出一个像神经网络方法一样的有限体积。一旦经过良好培训,网络方法就作为有限体积方案实施,因此是网状,它取决于网状。不同于传统的数字方法,我们的方法可以从明确的CFL限制中解脱出来,并可以适应任何时间步骤大小来演变解决方案。对于热量方程式来说,观测到第一级趋同顺序,错误与空间网状大小有关,但观测到与网状大小无关。基于细胞平均神经网络方法可以急剧演变不连续性,引入了几乎零的数值扩散。冲击波和稀有花动波波可以被非线性双曲保护法捕捉到。