Machine & Deep Learning methods for solving PDEs, in particular, for fluid simulations
18/01/2021, 11:00, en ligne via Teams
The purpose of this seminar is to talk about some approaches in the literature dedicated to solve PDEs by using Machine & Deep learning methods. These approaches are included principally under two frameworks, namely data driven solution and data driven discov- ery. The first framework consists of seeking the hidden solution u(x, t) given a fixed model parameters ? and the second one consists of seeking the parameters ? that best describe the observed training data. At the beginning of my presentation, I will talk about the Physics Informed Neural Networks PINNs which consists of formulating the PDEs, initial and boundary conditions as a loss function which have to be minimized in order to con- verge to the solution. Some examples of this method in addition to other ones from its extensions will be presented, more precisely, Hidden Fluid Mechanics HFM and DeepXDE. After that I will talk about another approach based on Machine Leaning method, more precisely, the regression forests method which allows to learn the fluid dynamic from a set of simulated videos. The third section, will focus on the continuous convolutional CConv method developed by some researchers from Intel Lab which allows as the previous one to learn the fluid dynamics from some simulated videos. Finally, I will talk about the Graph Network-based Simulators GNS, developed by DeepMind team, Google, which is in the same spirit as the two previous ones. This latter is more powerful in predicting complex fluid dynamics more accurately, with a powerful way which is difficult to distinguish from standard simulators.
Keywords
Data driven solution, Data driven discovery, Physics Informed Neural Net- works PINNs, Hidden Fluid Mechanics HFM, DeepXDE, Regression Forests, Continuous Convolutional CConv, Graph Network-based Simulators GNS.
References
[1] Ladicky ?, L., Jeong, S., Solenthaler, B., Pollefeys, M., and Gross, M., Data-driven fluid simu- lations using regression forests. ACM Trans. Graph. 34, 6, Article 199, November 2015.
[2] Raissi, M., Perdikaris, P., Em Karniadakis, G., Physics Informed Deep Learning (Part I): Data- driven Solutions of NonlinearPartial Differential Equations, arXiv:1711.10561, 2017.
[3] Lu, L., Meng, X., Mao, Z.,E. Karniadakis, G., DeepXDE: A deep learning library for solving differential equations, arXiv:1907.04502, 2020
[4] Raissi, M, Alireza Y., Em Karniadakis, G., Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data. arXiv, 1808.04327, 2018.
[5] Ummenhofer, B., Prantl, L., Thuerey, N., Koltun, V., Lagrangian fluid simulation with continu- ous convolutions. ICLR 2020 Conference, 2019.
[6] Sanchez-Gonzalez, A., Godwin, J., Pfaff, T., Ying, R., Leskovec J., Battaglia, P.W., Learning to Simulate Complex Physics with Graph Networks. The 37th International Conference on Machine Learning, Vienna, Austria, PMLR 119, 2020.
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