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This paper focuses on discussing Newton's method and its hybrid with machinelearning for the steady state Navier-Stokes Darcy model discretized by mixedelement methods. First, a Newton iterative method is introduced for solving therelative discretized problem. It is proved technically that this methodconverges quadratically with the convergence rate independent of the finiteelement mesh size, under certain standard conditions. Later on, a deep learningalgorithm is proposed for solving this nonlinear coupled problem. Following theideas of an earlier work by Huang, Wang and Yang (2020), an Int-Deep algorithmis constructed by combining the previous two methods so as to further improvethe computational efficiency and robustness. A series of numerical examples arereported to show the numerical performance of the proposed methods.

Newton's method and its hybrid with machine learning for Navier-Stokes Darcy Models discretized by mixed element methods