A Discontinuous Deep Ritz Method for PDEs
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We propose a novel deep learning approach for solving partial differential equations (PDEs), drawing inspiration from the classical discontinuous Galerkin method. Similar to the finite element method, we partition the domain and approximate the PDE solution within each element using polynomials of a specified degree. We construct a neural network that attempts to compute the solution’s degrees of freedom. We propose a loss function based on the quadratic Ritz functional, corrected with jump terms to enforce the solution and flux continuity on the mesh skeleton. In the optimization process, we employ exact (Gaussian) quadrature rules and automatic differentiation. In this talk, we will delve into the details of the method and its implementation. We will assess its efficiency by experimenting with different architectures, optimizers, and weights in the loss function with a simple 1D problem. Then, we will show how the proposed method performs with several two-dimensional boundary value problems with discontinuities in the data and/or the solution.
