In this paper, we study Physics-Informed Neural Networks (PINN) to approximate solutions to one-dimensional boundary value problems for linear elliptic equations and establish robust error estimates of PINN regardless of the quantities of the coefficients. In particular, we rigorously demonstrate the existence and uniqueness of solutions using the Sobolev space theory based on a variational approach. Deriving $L^2$-contraction estimates, we show that the error, defined as the mean square for differences at the sample points between the true solution and our trial function, is dominated by the training loss. Furthermore, we show that as the quantities of the coefficients for the differential equation increase, the error-to-loss ratio rapidly decreases. Our theoretical and experimental results confirm the robustness of the error regardless of the quantities of the coefficients.

翻译：本文研究物理信息神经网络（PINN）用于逼近线性椭圆方程一维边值问题的解，并建立了与系数大小无关的PINN鲁棒误差估计。特别地，我们基于变分方法，利用Sobolev空间理论严格证明了解的存在唯一性。通过推导$L^2$压缩估计，我们证明了真实解与试探函数在采样点处差值的均方误差由训练损失主导。此外，我们表明随着微分方程系数的增大，误差-损失比迅速下降。我们的理论与实验结果共同证实了误差估计相对于系数大小的鲁棒性。