  This paper focuses on discussing Newton's method and its hybrid with machine
learning for the steady state Navier-Stokes Darcy model discretized by mixed
element methods. First, a Newton iterative method is introduced for solving the
relative discretized problem. It is proved technically that this method
converges quadratically with the convergence rate independent of the finite
element mesh size, under certain standard conditions. Later on, a deep learning
algorithm is proposed for solving this nonlinear coupled problem. Following the
ideas of an earlier work by Huang, Wang and Yang (2020), an Int-Deep algorithm
is constructed by combining the previous two methods so as to further improve
the computational efficiency and robustness. A series of numerical examples are
reported to show the numerical performance of the proposed methods.
