  In this paper, we introduce cell-average based neural network (CANN) method
to solve high-dimensional parabolic partial differential equations. The method
is based on the integral or weak formulation of partial differential equations.
A feedforward network is considered to train the solution average of cells in
neighboring time. Initial values and approximate solution at $t=\Delta t$
obtained by high order numerical method are taken as the inputs and outputs of
network, respectively. We use supervised training combined with a simple
backpropagation algorithm to train the network parameters. We find that the
neural network has been trained to optimality for high-dimensional problems,
the CFL condition is not strictly limited for CANN method and the trained
network is used to solve the same problem with different initial values. For
the high-dimensional parabolic equations, the convergence is observed and the
errors are shown related to spatial mesh size but independent of time step
size.
