We analyze neural network solutions to partial differential equations obtained with Physics Informed Neural Networks. In particular, we apply tools of classical finite element error analysis to obtain conclusions about the error of the Deep Ritz method applied to the Laplace and the Stokes equations. Further, we develop an a posteriori error estimator for neural network approximations of partial differential equations. The proposed approach is based on the dual weighted residual estimator. It is destined to serve as a stopping criterion that guarantees the accuracy of the solution independently of the design of the neural network training. The result is equipped with computational examples for Laplace and Stokes problems.

翻译：我们分析通过物理知情神经网络获得的局部差异方程式的神经网络解决方案,特别是,我们运用传统限值元素错误分析工具,就Laplace和Stokes等式应用的Deep Ritz方法的错误作出结论;此外,我们开发了局部差异方程式神经网络近似的事后误差估计仪;拟议方法以双加权剩余估量仪为基础;该方法将作为一个停止标准,保证解决方案的准确性,而不受神经网络培训的设计影响;结果配有Laplace和Stoks问题的计算示例。