We introduce a Robust version of the Variational Physics-Informed Neural Networks method (RVPINNs). As in VPINNs, we define the quadratic loss functional in terms of a Petrov–Galerkin-type variational formulation of the PDE problem: the trial space is a (Deep) Neural Network (DNN) manifold, while the test space is a finite-dimensional vector space. Whereas the VPINN’s loss depends upon the selected basis functions of a given test space, herein, we minimize a loss based on the discrete dual norm of the residual. The main advantage of such a loss definition is that it provides a reliable and efficient estimator of the true error in the energy norm under the assumption of the existence of a local Fortin operator. We test the performance and robustness of our algorithm in several advection–diffusion problems. These numerical results perfectly align with our theoretical findings, showing that our estimates are sharp.

中文翻译：

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鲁棒的变分物理学神经网络

我们引入了变分物理信息神经网络方法 (RVPINN) 的鲁棒版本。与 VPINN 一样，我们根据 PDE 问题的 Petrov-Galerkin 型变分公式来定义二次损失函数：试验空间是（深度）神经网络 (DNN) 流形，而测试空间是有限维向量空间。尽管 VPINN 的损失取决于给定测试空间的所选基函数，但本文中，我们根据残差的离散对偶范数最小化损失。这种损失定义的主要优点是，在存在局部 Fortin 算子的假设下，它提供了能量范数真实误差的可靠且有效的估计器。我们在几个平流扩散问题中测试了我们的算法的性能和鲁棒性。这些数值结果与我们的理论结果完全一致，表明我们的估计是准确的。