Author:
Berrone Stefano,Canuto Claudio,Pintore Moreno
Abstract
AbstractWe consider the discretization of elliptic boundary-value problems by variational physics-informed neural networks (VPINNs), in which test functions are continuous, piecewise linear functions on a triangulation of the domain. We define an a posteriori error estimator, made of a residual-type term, a loss-function term, and data oscillation terms. We prove that the estimator is both reliable and efficient in controlling the energy norm of the error between the exact and VPINN solutions. Numerical results are in excellent agreement with the theoretical predictions.
Publisher
Springer Science and Business Media LLC
Reference22 articles.
1. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019). https://doi.org/10.1016/j.jcp.2018.10.045
2. Tartakovsky, A.M., Marrero, C.O., Perdikaris, P., Tartakovsky, G.D., Barajas-Solano, D.: Learning parameters and constitutive relationships with physics informed deep neural networks. arXiv preprint arXiv:1808.03398 (2018)
3. Yang, Y., Perdikaris, P.: Adversarial uncertainty quantification in physics-informed neural networks. J. Comput. Phys. 394, 136–152 (2019)
4. Lanthalet, S., Mishra, S., Karniadakis, G.E.: Error estimates for deeponets: a deep learning framework in infinite dimensions. arXiv preprint arXiv:2102.09618v2 (2021)
5. Kharazmi, E., Zhang, Z., Karniadakis, G.E.: VPINNs: variational physics-informed neural networks for solving partial differential equations. arXiv preprint arXiv:1912.00873 (2019)
Cited by
6 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献