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. 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 numerically and the results agree with the theoretical findings.

翻译：本文中，核近似被用作深度Ritz方法中的试探函数。该方法能够结合Nitsche方法弱施加边界条件，从而近似椭圆型偏微分方程的弱解。通过综合运用椭圆方程弱解的标准结论与核方法既有的收敛性结果，我们在不同范数下证明了先验误差估计。这种先验误差估计的可获得性使得该方法具有实际应用价值。文中详细描述了计算流程，同时提供了实践提示与实现细节。通过数值算例，我们对所提方法的性能进行了数值评估，其结果与理论结论一致。