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Variational Formulations of the Strong Formulation -- Forward and Inverse Modeling using Isogeometric Analysis and Physics-Informed Networks
Published 6 Dec 2023 in math.NA and cs.NA | (2312.03496v1)
Abstract: The recently introduced Physics-Informed Neural Networks (PINNs) have popularized least squares formulations of both forward and inverse problems involving partial differential equations (PDEs) in strong form. We employ both Isogeometric Analysis and Physics-Informed Networks.
- D. Braess. Stability of saddle point problems with penalty. ESAIM: Mathematical Modelling and Numerical Analysis, 30(6):731–742, 1996.
- J. H. Bramble and S. Hilbert. Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM Journal on Numerical Analysis, 7(1):112–124, 1970.
- F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Publications des séminaires de mathématiques et informatique de Rennes, (S4):1–26, 1974.
- Isogeometric analysis: toward integration of CAD and FEA. John Wiley & Sons, 2009.
- M. Dissanayake and N. Phan-Thien. Neural-network-based approximations for solving partial differential equations. Communications in Numerical Methods in Engineering, 10(3):195–201, 1994.
- P. Grisvard. Elliptic problems in nonsmooth domains. SIAM, 2011.
- A new framework for the stability analysis of perturbed saddle-point problems and applications in poromechanics. Mathematics of Computation, 2023.
- Physics-informed machine learning. Nature Reviews Physics, 3(6):422–440, 2021.
- Robust preconditioners for PDE-constrained optimization with limited observations. BIT Numerical Mathematics, 57:405–431, 2017.
- Robust preconditioning and error estimates for optimal control of the convection–diffusion–reaction equation with limited observation in isogeometric analysis. SIAM Journal on Numerical Analysis, 60(1):195–221, 2022.
- L. McClenny and U. Braga-Neto. Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv preprint arXiv:2009.04544, 2020.
- J. Müller and M. Zeinhofer. Achieving high accuracy with PINNs via energy natural gradient descent. In A. Krause, E. Brunskill, K. Cho, B. Engelhardt, S. Sabato, and J. Scarlett, editors, Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pages 25471–25485. PMLR, 23–29 Jul 2023. URL https://proceedings.mlr.press/v202/muller23b.html.
- Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378:686–707, 2019.
- J. Sogn and W. Zulehner. Schur complement preconditioners for multiple saddle point problems of block tridiagonal form with application to optimization problems. IMA Journal of Numerical Analysis, 39(3):1328–1359, 2019.
- Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM Journal on Scientific Computing, 43(5):A3055–A3081, 2021.
- When and why pinns fail to train: A neural tangent kernel perspective. Journal of Computational Physics, 449:110768, 2022.
- A unified framework for the error analysis of physics-informed neural networks. arXiv preprint arXiv:2311.00529, 2023.
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