  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.
