Error Estimates for the Deep Ritz Method with Boundary Penalty

Abstract

We establish estimates on the error made by the Ritz method for quadratic energies on the space H 1 (Ω) in the approximation of the solution of variational problems with different boundary conditions. Special attention is paid to the case of Dirichlet boundary values which are treated with the boundary penalty method. We consider arbitrary and in general non linear classes V ⊆ H 1 (Ω) of ansatz functions and estimate the error in dependence of the optimisation accuracy, the approximation capabilities of the ansatz class and -in the case of Dirichlet boundary values -the penalisation strength λ. For non-essential boundary conditions the error of the Ritz method decays with the same rate as the approximation rate of the ansatz classes. For the boundary penalty method we obtain that given an approximation rate of r in H 1 (Ω) and an approximation rate of s in L 2 (∂Ω) of the ansatz classes, the optimal decay rate of the estimated error is min(s/2, r) ∈ [r/2, r] and achieved by choosing λn ∼ n s . We discuss how this rate can be improved, the relation to existing estimates for finite element functions as well as the implications for ansatz classes which are given through ReLU networks. Finally, we use the notion of Γ-convergence to show that the Ritz method converges for a wide class of energies including nonlinear stationary PDEs like the p-Laplace.

Error Analysis of Deep Ritz Methods for Elliptic Equations

“…Recently, [32,31] also study the convergence of DRM with Dirichlet boundary condition via penalty method. However, the analysis in [32,31] is based on some additional conditions, and we do not need these conditions to obtain the error inducing by the penalty.…”

“…Recently, [32,31] also study the convergence of DRM with Dirichlet boundary condition via penalty method. However, the analysis in [32,31] is based on some additional conditions, and we do not need these conditions to obtain the error inducing by the penalty. More importantly, we provide the convergence rate analysis involving the statistical error caused by finite samples used in the SGD training, while in [32,31] they do not consider the statistical error at all.…”

“…several years, a challenging and interesting question is to provide a rigorous error analysis such as finite element method. Several recent efforts have been devoted to making processes along this line, see for example [11,28,30,32,26,20,36,41,10]. In [28], least squares minimization method with two-layer neural networks is studied, the optimization error under the assumption of over-parametrization and generalization error without the over-parametrization assumption are analyzed.…”

Deep Ritz method for the spectral fractional Laplacian equation using the Caffarelli-Silvestre extension

“…Clearly, S 2,M,ReLU,h is a subset of N 2,M,ReLU,h . The following theorem and proof are referred to the result in [23] for deep Ritz method in a bounded domain. Theorem 3.1.…”

Generalization Error Analysis of Neural networks with Gradient Based Regularization

“…Instead, we can analyze the generalization error. Some generalization error analysis have been developed for some PDE-related problems [22,23,24,25], but the study of the generalization error for general variational problems is still very few, especially for total variation related problems.…”