A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions (Communications in Mathematical Research, 2020)
Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensionalcase. Unlike classical numerical methods, such as finite difference method andfinite element method, the enforcement of boundary conditions in deep neural networks is highly nontrivial. One general strategy is to use the penaltymethod. In the work, we conduct a comparison study for elliptic problems withfour different boundary conditions, i.e., Dirichlet, Neumann, Robin, and periodic boundary conditions, using two representative methods: deep Galerkinmethod and deep Ritz method. In the former, the PDE residual is minimizedin the least-squares sense while the corresponding variational problem is minimized in the latter. Therefore, it is reasonably expected that deep Galerkinmethod works better for smooth solutions while deep Ritz method works better for low-regularity solutions. However, by a number of examples, we observe that deep Ritz method can outperform deep Galerkin method with a cleardependence of dimensionality even for smooth solutions and deep Galerkinmethod can also outperform deep Ritz method for low-regularity solutions.Besides, in some cases, when the boundary condition can be implemented inan exact manner, we find that such a strategy not only provides a better approximate solution but also facilitates the training process.
Keywords: Partial differential equations; boundary conditions; deep Galerkin method; deep Ritz method; penalty method.