Author:
Berrone Stefano,Canuto Claudio,Pintore Moreno
Abstract
AbstractIn this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the importance of the inf-sup condition. Our results suggest, somehow counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of suitably high precision.
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,General Engineering,Theoretical Computer Science,Software,Applied Mathematics,Computational Mathematics,Numerical Analysis
Reference36 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:1808.03398 (2018)
3. Yang, Y., Perdikaris, P.: Adversarial uncertainty quantification in physics-informed neural networks. J. Comput. Phys. 394, 136–152 (2019)
4. Lanthaler, S., Mishra, S., Karniadakis, G.E.: Error estimates for DeepONets: a deep learning framework in infinite dimensions. Trans. Math. Appl. 6(1), tnac001 (2022). https://doi.org/10.1093/imatrm/tnac001
5. Elbrächter, D., Perekrestenko, D., Grohs, P., Bölcskei, H.: Deep neural network approximation theory. IEEE Trans. Inf. Theory 67(5), 2581–2623 (2021)
Cited by
19 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