Kernel methods in the deep Ritz framework: theory and practice

In this contribution, kernel approximations are applied as ansatz functions within the Deep Ritz method. This allows to approximate weak solutions of elliptic partial differential equations with weak enforcement of boundary conditions using Nitsche’s method. A priori error estimates are proven in different norms leveraging both standard results for weak solutions of elliptic equations and well-established convergence results for kernel methods. This availability of a priori error estimates renders the method useful for practical purposes where theoretical guarantees are important. The procedure is described in detail, meanwhile providing practical hints and implementation details. By means of numerical examples, the performance of the proposed approach is evaluated and the theoretical findings are verified. We further apply two-layered kernels in a high-dimensional test case, which goes beyond a purely linear method. To the best of our knowledge, this paper contains the first extensive numerical studies and discussions comparing kernel interpolation, the kernel Galerkin method, the deep Ritz method with kernels and the “classical” deep Ritz method with neural networks. We numerically verify advantages and disadvantages of the particular methods and show under which assumptions the individual approaches are beneficial.