We establish estimates on the error made by the Deep Ritz Method for elliptic problems on the space $H^1(\Omega)$ with different boundary conditions. For Dirichlet boundary conditions, we estimate the error when the boundary values are approximately enforced through the boundary penalty method. Our results apply to arbitrary and in general non linear classes $V\subseteq H^1(\Omega)$ of ansatz functions and estimate the error in dependence of the optimization accuracy, the approximation capabilities of the ansatz class and -- in the case of Dirichlet boundary values -- the penalisation strength $\lambda$. 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 essential boundary conditions, given an approximation rate of $r$ in $H^1(\Omega)$ and an approximation rate of $s$ in $L^2(\partial\Omega)$ of the ansatz classes, the optimal decay rate of the estimated error is $\min(s/2, r)$ and achieved by choosing $\lambda_n\sim n^{s}$. We discuss the implications for ansatz classes which are given through ReLU networks and the relation to existing estimates for finite element functions.
翻译:我们确定深利兹方法对具有不同边界条件的空域的椭圆问题的误差的估计数。 对于迪里赫莱特边界条件,我们估计边界值通过边界处罚方法大致执行时的误差。我们的结果适用于任意和非线性等级的误差和一般非线性等级的误差。 我们的结果适用于安萨兹函数的任意和一般非线性类别 $V\subseteq H%1(奥米加),并估计优化精确度依赖度的误差、安萨兹级的近距离能力以及 -- -- 就迪里赫莱特边界值而言 -- -- 估计差差差差值的坏坏率是 $2 / lambda 值。 对于非必要的边界条件,我们根据1美元(奥米加)的近差率和亚萨兹级的美元近差率,估计差的最佳坏坏率是 $/2, r) rbda 。我们通过目前等级选择的兰基值 网络 和Rembxx 来得出一个影响。