  In theory, boundary and initial conditions are important for the
wellposedness of partial differential equations (PDEs). Numerically, these
conditions can be enforced exactly in classical numerical methods, such as
finite difference method and finite element method. Recent years have witnessed
growing interests in solving PDEs by deep neural networks (DNNs), especially in
the high-dimensional case. However, in the generic situation, a careful
literature review shows that boundary conditions cannot be enforced exactly for
DNNs, which inevitably leads to a modeling error. In this work, based on the
recently developed deep mixed residual method (MIM), we demonstrate how to make
DNNs satisfy boundary and initial conditions automatically in a systematic
manner. As a consequence, the loss function in MIM is free of the penalty term
and does not have any modeling error. Using numerous examples, including
Dirichlet, Neumann, mixed, Robin, and periodic boundary conditions for elliptic
equations, and initial conditions for parabolic and hyperbolic equations, we
show that enforcing exact boundary and initial conditions not only provides a
better approximate solution but also facilitates the training process.
