  There has been an arising trend of adopting deep learning methods to study
partial differential equations (PDEs). This article is to propose a Deep
Learning Galerkin Method (DGM) for the closed-loop geothermal system, which is
a new coupled multi-physics PDEs and mainly consists of a framework of
underground heat exchange pipelines to extract the geothermal heat from the
geothermal reservoir. This method is a natural combination of Galerkin Method
and machine learning with the solution approximated by a neural network instead
of a linear combination of basis functions. We train the neural network by
randomly sampling the spatiotemporal points and minimize loss function to
satisfy the differential operators, initial condition, boundary and interface
conditions. Moreover, the approximate ability of the neural network is proved
by the convergence of the loss function and the convergence of the neural
network to the exact solution in L^2 norm under certain conditions. Finally,
some numerical examples are carried out to demonstrate the approximation
ability of the neural networks intuitively.
