Singular layer Physics Informed Neural Network method for Plane Parallel Flows (2311.15304v1)
Abstract: We construct in this article the semi-analytic Physics Informed Neural Networks (PINNs), called {\em singular layer PINNs} (or {\em sl-PINNs}), that are suitable to predict the stiff solutions of plane-parallel flows at a small viscosity. Recalling the boundary layer analysis, we first find the corrector for the problem which describes the singular behavior of the viscous flow inside boundary layers. Then, using the components of the corrector and its curl, we build our new {\em sl-PINN} predictions for the velocity and the vorticity by either embedding the explicit expression of the corrector (or its curl) in the structure of PINNs or by training the implicit parts of the corrector (or its curl) together with the PINN predictions. Numerical experiments confirm that our new {\em sl-PINNs} produce stable and accurate predicted solutions for the plane-parallel flows at a small viscosity.
- Theory-guided physics-informed neural networks for boundary layer problems with singular perturbation. J. Comput. Phys. 473 (2023), Paper No. 111768, 15 pp.
- Physics-informed neural networks and functional interpolation for stiff chemical kinetics. Chaos 32, 063107 (2022).
- Olga Fuks and Hamdi A Tchelepi. Limitations of physics informed machine learning for nonlinear two-phase transport in porous media. Journal of Machine Learning for Modeling and Computing, 1(1), 2020.
- Singular perturbations and boundary layers, volume 200 of Applied Mathematical Sciences. Springer Nature Switzerland AG, 2018. https://doi.org/10.1007/978-3-030-00638-9
- The vanishing viscosity limit for some symmetric flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 5, pp. 1237–1280, DOI 10.1016/J.ANIHPC.2018.11.006
- Antonio Tadeu Azevedo Gomes and Larissa Miguez da Silva and Frederic Valentin. Physics-Aware Neural Networks for Boundary Layer Linear Problems. arXiv preprint, 2022.
- On the numerical approximations of stiff convection-diffusion equations in a circle. Numer. Math., 127(2):291–313, 2014.
- NSFnets (Navier– Stokes flow nets): Physics–informed neural networks for the incompressible Navier– Stokes equations. Journal of Computational Physics. 426 (2021), 109951.
- Deep Learning in Computational Mechanics. Studies in Computational Intelligence, Springer International Publishing, Cham (2021). https://doi.org/10.1007/978-3-030-76587-3
- Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw., 1998;9(5):987-1000.
- Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys., 378 (2019), pp. 686–707.
- Numerical methods for singularly perturbed differential equations, volume 24 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1996.
- Martin Stynes. Steady-state convection-diffusion problems. Acta Numer., 14:445–508, 2005.
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