Основні поняття
Robust Variational Physics-Informed Neural Networks provide reliable error estimations in energy norms.
Анотація
The article introduces Robust Variational Physics-Informed Neural Networks (RVPINNs) as a method to minimize errors in energy norms. It discusses the limitations of previous methods like PINNs and VPINNs and proposes a new approach. The core idea is to minimize the residual in the discrete dual norm, providing a reliable estimator of the true error. The methodology is tested in various advection-diffusion problems, showcasing its robustness. The article is structured into sections covering Introduction, Preliminaries, Robust Variational Physics-Informed Neural Networks, Error estimates for RVPINNs, Numerical examples, and more.
Статистика
"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."
"The Gram matrix becomes the identity when considering an orthonormal discrete basis with respect to the inner product in the test space."
"The norm of the true error is bounded from below and above by the norm of the residual representative."
Цитати
"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."
"The Gram matrix becomes the identity when considering an orthonormal discrete basis with respect to the inner product in the test space."
"The norm of the true error is bounded from below and above by the norm of the residual representative."