We introduce a Robust version of the Physics-Informed Neural Networks (RPINNs) to approximate the Partial Differential Equations (PDEs) solution. Standard Physics Informed Neural Networks (PINN) takes into account the governing physical laws described by PDE during the learning process. The network is trained on a data set that consists of randomly selected points in the physical domain and its boundary. PINNs have been successfully applied to solve various problems described by PDEs with boundary conditions. The loss function in traditional PINNs is based on the strong residuals of the PDEs. This loss function in PINNs is generally not robust with respect to the true error. The loss function in PINNs can be far from the true error, which makes the training process more difficult. In particular, we do not know if the training process has already converged to the solution with the required accuracy. This is especially true if we do not know the exact solution, so we cannot estimate the true error during the training. This paper introduces a different way of defining the loss function. It incorporates the residual and the inverse of the Gram matrix, computed using the energy norm. We test our RPINN algorithm on two Laplace problems and one advection-diffusion problem in two spatial dimensions. We conclude that RPINN is a robust method. The proposed loss coincides well with the true error of the solution, as measured in the energy norm. Thus, we know if our training process goes well, and we know when to stop the training to obtain the neural network approximation of the solution of the PDE with the true error of required accuracy.

翻译：我们提出了一种鲁棒物理信息神经网络（RPINNs），用于逼近偏微分方程（PDEs）的解。标准物理信息神经网络（PINN）在学习过程中考虑了由偏微分方程描述的控制物理定律。该网络是在包含物理域及其边界上随机选取点的数据集上进行训练的。PINN已成功应用于求解各种带边界条件的偏微分方程问题。传统PINN中的损失函数基于PDE的强残差，这种损失函数通常对真实误差缺乏鲁棒性。PINN的损失函数可能与真实误差相差甚远，导致训练过程更加困难。尤其当无法获知精确解时，我们无法在训练过程中估计真实误差，因此难以判断训练是否已收敛到所需精度的解。本文提出了一种定义损失函数的新方法：该方法结合了残差与通过能量范数计算的Gram矩阵逆。我们通过在两个拉普拉斯问题和一个二维空间对流扩散问题上的测试，验证了RPINN算法的鲁棒性。实验证明，所提出的损失函数与以能量范数衡量的解的真实误差具有良好的一致性。因此，我们能够准确判断训练过程的收敛状态，并确定何时终止训练，从而获得满足所需真实误差精度的偏微分方程解的神经网络近似。