  Many problems in science and engineering can be represented by a set of
partial differential equations (PDEs) through mathematical modeling.
Mechanism-based computation following PDEs has long been an essential paradigm
for studying topics such as computational fluid dynamics, multiphysics
simulation, molecular dynamics, or even dynamical systems. It is a vibrant
multi-disciplinary field of increasing importance and with extraordinary
potential. At the same time, solving PDEs efficiently has been a long-standing
challenge. Generally, except for a few differential equations for which
analytical solutions are directly available, many more equations must rely on
numerical approaches such as the finite difference method, finite element
method, finite volume method, and boundary element method to be solved
approximately. These numerical methods usually divide a continuous problem
domain into discrete points and then concentrate on solving the system at each
of those points. Though the effectiveness of these traditional numerical
methods, the vast number of iterative operations accompanying each step forward
significantly reduces the efficiency. Recently, another equally important
paradigm, data-based computation represented by deep learning, has emerged as
an effective means of solving PDEs. Surprisingly, a comprehensive review for
this interesting subfield is still lacking. This survey aims to categorize and
review the current progress on Deep Neural Networks (DNNs) for PDEs. We discuss
the literature published in this subfield over the past decades and present
them in a common taxonomy, followed by an overview and classification of
applications of these related methods in scientific research and engineering
scenarios. The origin, developing history, character, sort, as well as the
future trends in each potential direction of this subfield are also introduced.
