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.

翻译：本文重点讨论了牛顿法及其与机器学习结合的混合方法，用于混合单元法离散的稳态纳维-斯托克斯-达西模型。首先，引入牛顿迭代法求解相应的离散问题。通过严格证明，在特定标准条件下，该方法具有二次收敛性，且收敛速度与有限元网格尺寸无关。随后，提出了用于求解该非线性耦合问题的深度学习算法。借鉴Huang、Wang与Yang（2020）前期工作的思路，将前两种方法结合构建了Int-Deep算法，以进一步提升计算效率与鲁棒性。最后，通过一系列数值算例展示了所提方法的数值性能。