Efficient coupled deep neural networks for the time-dependent coupled Stokes-Darcy problems

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• Our method compiles the interface conditions of the coupled PDEs into the networks properly. • This method can be served as an efficient alternative to the complex coupled problems. • We sampled randomly and only input spatial coordinates without being restricted by the nature of samples. • Our method is meshfree and parallel which can solve multiple variables independently at the same time. • We give the theory to guarantee the convergence of the loss function and the convergence of the neural networks to the exact solution. • We present the numerical examples in both 2D and 3D cases. In this paper, we propose and investigate an efficient method called CDNNs (Coupled Deep Neural Networks) for the time-dependent coupled Stokes-Darcy problems. Specifically, we encode complex interface conditions related to the variables of the coupled problems into several neural networks to constrain the approximation solution. We define a custom loss function to guarantee the physical properties of the numerical solution as well as the conservation of the energy. In particular, the present method is mesh-free since it only inputs random spatiotemporal points and can avoid the difficulties and complexities caused by the mesh-based method. Moreover, our method is parallel, it solves each variable simultaneously and independently. Furthermore, we obtain the convergence analysis to illustrate the capabilities of our method for solving the coupled problems. Numerical experiments further demonstrate the accuracy and efficiency of the proposed method.