Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions

12/03/2019

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by   Yulei Liao, et al.

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We propose a method due to Nitsche (Deep Nitsche Method) from 1970s to deal with the essential boundary conditions encountered in the deep learning-based numerical method without significant extra computational costs. The method inherits several advantages from Deep Ritz Method <cit.> while successfully overcomes the difficulties in treatment of the essential boundary conditions. We illustrate the method on several representative problems posed in at most 100 dimensions with complicated boundary conditions. The numerical results clearly show that the Deep Nitsche Method is naturally nonlinear, naturally adaptive and has the potential to work on rather high dimensions.