Robust Variational Physics-Informed Neural Networks

• Published in: Computer methods in applied mechanics and engineering Vol. 425; p. 116904
• Main Authors: Rojas, Sergio, Maczuga, Paweł, Muñoz-Matute, Judit, Pardo, David, Paszyński, Maciej
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.05.2024
• Subjects: A posteriori error estimation; Minimum residual principle; Petrov–Galerkin formulation; Riesz representation; Robustness; Variational Physics-Informed Neural Networks; Variational Physics-Informed Neural Networks; Minimum residual principle; Robustness; Petrov–Galerkin formulation; A posteriori error estimation; Riesz representation
• ISSN: 0045-7825; 1879-2138; 1879-2138
• DOI: 10.1016/j.cma.2024.116904

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. •We define robust loss functionals for Variational Physics Informed Neural Networks.•We prove that the proposed loss functional is equivalent to the true error up to an oscillation term.•Unlike classical VPINNs, our approach is not sensitive to the choice of the basis functions in the test space.•We test our strategy in several 1D and 2D elliptic boundary-value problems, showing the robustness of the approach.