  Deep neural networks are powerful tools for approximating functions, and they
are applied to successfully solve various problems in many fields. In this
paper, we propose a neural network-based numerical method to solve partial
differential equations. In this new framework, the method is designed on weak
formulations, and the unknown functions are approximated by deep neural
networks and test functions can be chosen by different approaches, for
instance, basis functions of finite element methods, neural networks, and so
on. Because the spaces of trial function and test function are different, we
name this new approach by Deep Petrov-Galerkin Method (DPGM). The resulted
linear system is not necessarily to be symmetric and square, so the discretized
problem is solved by a least-square method. Take the Poisson problem as an
example, mixed DPGMs based on several mixed formulations are proposed and
studied as well. In addition, we apply the DPGM to solve two classical
time-dependent problems based on the space-time approach, that is, the unknown
function is approximated by a neural network, in which temporal variable and
spatial variables are treated equally, and the initial conditions are regarded
as boundary conditions for the space-time domain. Finally, several numerical
examples are presented to show the performance of the DPGMs, and we observe
that this new method outperforms traditional numerical methods in several
aspects.
