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This paper proposes a deep-learning-based domain decomposition method (DeepDDM), which leverages deep neural networks (DNN) to discretize the subproblems divided by domain decomposition methods (DDM) for solving partial differential equations (PDE). Using DNN to solve PDE is a physics-informed learning problem with the objective involving two terms, domain term and boundary term, which respectively make the desired solution satisfy the PDE and corresponding boundary conditions. DeepDDM will exchange the subproblem information across the interface in DDM by adjusting the boundary term for solving each subproblem by DNN. Benefiting from the simple implementation and mesh-free strategy of using DNN for PDE, DeepDDM will simplify the implementation of DDM and make DDM more flexible for complex PDE, e.g., those with complex interfaces in the computational domain. This paper will firstly investigate the performance of using DeepDDM for elliptic problems, including a model problem and an interface problem. The numerical examples demonstrate that DeepDDM exhibits behaviors consistent with conventional DDM: the number of iterations by DeepDDM is independent of network architecture and decreases with increasing overlapping size. The performance of DeepDDM on elliptic problems will encourage us to further investigate its performance for other kinds of PDE and may provide new insights for improving the PDE solver by deep learning.
In this paper, we propose a deep unfitted Nitsche method for computing elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy minimi
We introduce an adaptive element-based domain decomposition (DD) method for solving saddle point problems defined as a block two by two matrix. The algorithm does not require any knowledge of the constrained space. We assume that all sub matrices are
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We design an adaptive unfitted finite element method on the Cartesian mesh with hanging nodes. We derive an hp-reliable and efficient residual type a posteriori error estimate on K-meshes. A key ingredient is a novel hp-domain inverse estimate which