We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet boundary conditions, we estimate the error when the boundary values are imposed through the boundary penalty method. Our results apply to arbitrary sets 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 penalization strength $\lambda$. To the best of our knowledge, our results are presently the only ones in the literature that treat the case of Dirichlet boundary conditions in full generality, i.e., without a lower order term that leads to coercivity on all of $H^1(\Omega)$. Further, we discuss the implications of our results for ansatz classes which are given through ReLU networks and the relation to existing estimates for finite element functions. For high dimensional problems our results show that the favourable approximation capabilities of neural networks for smooth functions are inherited by the Deep Ritz Method.
翻译:我们估计了线性椭圆方程式的深Ritz方法错误。 对于 Dirichlet 边界条件, 我们估计了边界值通过边界处罚方法强制时的错误。 我们的结果适用于任意的 ansatz 函数组, 并估计了取决于优化准确度、 ansatz 类近似能力的错误, 以及( 就 Dirichlet 边界值而言) 的惩罚强度 。 据我们所知, 我们的结果目前是唯一在文献中完全笼统地处理 Dirichlet 边界条件案例的错误, 也就是说, 没有更低的顺序, 导致所有H1 (\\ Omega) $ 的共振动。 此外, 我们讨论我们的结果对通过 RELU 网络提供的 ansatz 类的影响, 以及与现有有限元素函数估计的关系。 关于高维问题, 我们的结果表明, 深利茨 方法 继承了 。