A deep learning initialized iterative (Int-Deep) method is developed for numerically solving Navier-Stokes Darcy model. For this purpose, Newton iterative method is mentioned for solving the relative finite element discretized problem. It is proved 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 for the previous problem in order to further improve the computational efficiency. A series of numerical examples are reported to confirm that the Int-Deep algorithm converges to the true solution rapidly and is robust with respect to the physical parameters in the model.

翻译：针对Navier-Stokes Darcy模型的数值求解，本文发展了一种深度学习初始化迭代（Int-Deep）方法。该方法首先引入牛顿迭代法求解相关的有限元离散问题，并证明在特定标准条件下，该迭代法具有与有限元网格尺寸无关的二次收敛率。随后，针对该非线性耦合问题，提出一种深度学习算法。借鉴Huang、Wang与Yang（2020）的早期工作思想，本文构建了面向该问题的Int-Deep算法，以进一步提升计算效率。一系列数值实验证实，Int-Deep算法能够快速收敛至真实解，且对模型中的物理参数具有鲁棒性。