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.

翻译：我们提出了鲁棒变分物理信息神经网络方法（RVPINNs）。与VPINNs类似，我们基于PDE问题的Petrov-Galerkin型变分公式定义二次损失泛函：试验空间为（深度）神经网络（DNN）流形，而测试空间为有限维向量空间。尽管VPINN的损失依赖于给定测试空间中选取的基函数，但本文我们基于残差的离散对偶范数最小化损失。这种损失定义的主要优势在于，在假设存在局部Fortin算子的条件下，它能提供能量范数下真实误差的可靠且高效的估计量。我们在多个对流扩散问题中测试了算法的性能和鲁棒性。这些数值结果与理论分析完美吻合，表明我们的估计是精确的。