Abstract
Data assimilation (DA) refers to methodologies which combine data and underlying governing equations to provide an estimation of a complex system. Physics informed neural network (PINN) provides an innovative machine learning technique for solving and discovering the physics in nature. By encoding general nonlinear partial differential equations, which govern different physical systems such as fluid flows, to the deep neural network, PINN can be used as a tool for DA. Due to its nature that neither numerical differential operation nor temporal and spatial discretization is needed, PINN is straightforward for implementation and getting more and more attention in the academia. In this paper, we apply the PINN to several flow problems and explore its potential in fluid physics. Both the mesoscopic Boltzmann equation and the macroscopic Navier-Stokes are considered as physics constraints. We first introduce a discrete Boltzmann equation informed neural network and evaluate it with a one-dimensional propagating wave and two-dimensional lid-driven cavity flow. Such laminar cavity flow is also considered as an example in an incompressible Navier-Stokes equation informed neural network. With parameterized Navier-Stokes, two turbulent flows, one within a C-shape duct and one passing a bump, are studied and accompanying pressure field is obtained. Those examples end with a flow passing through a porous media. Applications in this paper show that PINN provides a new way for intelligent flow inference and identification, ranging from mesoscopic scale to macroscopic scale, and from laminar flow to turbulent flow.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Zhang Z., Moore J. C. Mathematical and physical fundamentals of climate change [M]. Boston, USA: Elsevier, 2015.
Michelén Ströfer C. A., Zhang X., Xiao H. et al. Enforcing boundary conditions on physical fields in Bayesian inversion [J]. Computer Methods in Applied Mechanics and Engineering, 2020, 367: 113097.
Meldi M., Poux A. A reduced order model based on Kalman filtering for sequential data assimilation of turbulent flows [J]. Journal of Computational Physics, 2017, 347: 207–234.
Verma S., Novati G., Koumoutsakos P. Efficient collective swimming by harnessing vortices through deep reinforcement learning [J]. Proceedings of the National Academy of Sciences of the United States of America, 2018, 115(23): 5849–5854.
Schmid P. J. Dynamic mode decomposition of numerical and experimental data [J]. Journal of Fluid Mechanics, 2010, 656: 5–28.
Bai X., Zhang W., Guo A. et al. The flip-flopping wake pattern behind two side-by-side circular cylinders: A global stability analysis [J]. Physics of Fluids, 2016, 28(4): 044102.
Bai X., Zhang W., Wang Y. Deflected oscillatory wake pattern behind two side-by-side circular cylinders [J]. Ocean Engineering, 2020, 197: 106847.
Brunton S. L., Noack B. R., Koumoutsakos P. Machine learning for fluid mechanics [J]. Annual Review of Fluid Mechanics, 2020, 52(1): 477–508.
Tracey B., Duraisamy K., Alonso J. Application of supervised learning to quantify uncertainties in turbulence and combustion modeling [C]. 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Texas, USA, 2013.
Tracey B., Duraisamy K., Alonso J. A machine learning strategy to assist turbulence model development [C]. 53rd AIAA Aerospace Sciences Meeting, Kissimmee, Florida, USA, 2015.
Zhang Z., Song X. D., Ye S. R. et al. Application of deep learning method to Reynolds stress models of channel flow based on reduced-order modeling of DNS data [J]. Journal of Hydrodynamics, 2019, 31(1): 58–65.
Ling J., Kurzawski A., Templeton J. Reynolds averaged turbulence modelling using deep neural networks with embedded invariance [J]. Journal of Fluid Mechanics, 2016, 807: 155–166.
Weatheritt J., Sandberg R. D. The development of algebraic stress models using a novel evolutionary algorithm [J]. International Journal of Heat and Fluid Flow, 2017, 68: 298–318.
Zhang X., Wu J., Coutier-Delgosha O. et al. Recent progress in augmenting turbulence models with physics-informed machine learning [J]. Journal of Hydrodynamics, 2019, 31(6): 1153–1158.
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]. Journal of Computational Physics, 2019, 378: 686–707.
Jin X., Cai S., Li H. et al. NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations [J]. Journal of Computational Physics, 2020(in Press).
Raissi M., Wang Z., Triantafyllou M. S. et al. Deep learning of vortex-induced vibrations [J]. Journal of Fluid Mechanics, 2019, 861: 119–137.
Raissi M., Yazdani A., Karniadakis G. E. Hidden fluid mechanics: Learning velocity and pressure fields from visualizations [J]. Science, 2020, 367(6481): 1026–1030.
Shukla K., Di Leoni P. C., Blackshire J. et al. Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks [J]. Journal of Nondestructive Evaluation, 2020, 39(3): 61.
Xu H., Zhang W., Wang Y. Explore missing flow dynamics by physics-informed deep learning: The parameterised governing systems [R]. 2020, arXiv: 2008.12266.
Lou Q., Meng X., Karniadakis G. Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation [R]. 2020, arXiv: 2010.09147.
Wang S., Teng Y., Perdikaris P. Understanding and mitigating gradient pathologies in physics-informed neural networks [R]. 2020, arXiv: 2001.04536.
He Y. L., Wang Y., Li Q. Lattice Boltzmann method: Theory and applications [M]. Beijing, China: Science Press, 2009.
Wang Y., He Y. L., Tang G. H. et al. Simulation of two-dimensional oscillating flow using the lattice Boltzmann method [J]. International Journal of Modern Physics C, 2006, 17(5): 615–630.
Xia Y. X., Qian Y. H. Lattice Boltzmann method for Casimir invariant of two-dimensional turbulence [J]. Journal of Hydrodynamics, 2016, 28(2): 319–324.
Abadi M., Barham P., Chen J. et al. TensorFlow: A system for large-scale machine learning [C]. Proceedings of the 12th USENIX conference on Operating Systems Design and Implementation, Savannah, GA, USA, 2016.
Li L., Mei R., Klausner J. F. Lattice Boltzmann models for the convection-diffusion equation: D2Q5 vs D2Q9 [J]. International Journal of Heat and Mass Transfer, 2017, 108: 41–62.
Haukur E. H. Porous media in OpenFOAM [R]. Gothenburg, Sweden: Chalmers University of Technology, 2009.
Acknowledgement
This work was supported by the Fundamental Research Funds for the Central Universities of China (Grant No. B200202049).
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (Grant Nos. 91851127, 51809084).
Biography: Xiao-dong Bai (1986-), Male, Ph. D.
Rights and permissions
About this article
Cite this article
Bai, Xd., Wang, Y. & Zhang, W. Applying physics informed neural network for flow data assimilation. J Hydrodyn 32, 1050–1058 (2020). https://doi.org/10.1007/s42241-020-0077-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42241-020-0077-2