本文結合變分自動編碼器(Variational AutoEncoder)以非監督式機器學習的方式，學習與預測流體在特定的速度場中與其相應的壓力場分布，並以此自動編碼器預測的壓力作為解壓力投影法(the pressure projection method)的初始條件，以加速壓力投影法在特定網格及邊界條件下的單板驅動空腔流的計算。本文提出的方法包括以變形的壓力帕松方程式(the Poisson equation of pressure) 新定義的速度場分布作為自動編碼器的輸入條件、特殊設計的變分自動編碼器架構。此編碼器可使預測重組的壓力場和原始速度場高度還原，而使直接預測的壓力初始條件有較高的準確性，以此達到加速計算的效果。此外，本文也透過設定不同的損失函數(loss function)使自動編碼器在學習後可預測更加準確的壓力分佈。為評估此自動編碼器的表現，本文分別探討從雷諾數(Reynolds number)為50到雷諾數為2000的情況下，以固定或變化的拉板速度拉動二維空腔上板時，自動編碼器預測壓力的準確性與加速效果。評估編碼器表現的標準為比較用預測的壓力作為初始條件及傳統計算流體力學使用的初始條件，兩者在特定疊代法(如雅可比或高斯-賽德爾疊代法)中達到滿足壓力帕松方程式的指定收斂條件時，所需要的疊代次數。就本文探討的流場來看，在預測未學習過的流場壓力時，此自動編碼器平均可以減少數千次的疊代步數，約可增加零點二到一倍的計算速度。並且，此自動編碼器預測的壓力在應用於多種疊代法、任意的時間步長甚至週期性拉板的計算時都有明顯的加速效果。

This paper combines an unsupervised machine learning model, Variational AutoEncoder (VAE), to accelerate the calculation of the pressure projection by predicting a more accurate initial pressure state for two-dimensional lid-driven cavity cases under domain grid sizes. The method consists of a numerical pre-processing to identify the input of velocity based on the Poisson equation of pressure, a developed VAE model, which reforms the fluid fields highly similar to the origin inputs, as a pressure projection solver to directly predict the initial pressure for the intermediate velocity input and a set loss function definition to lead the VAE model to better fit the ideal pressure output. The VAE model is validated by computing the steady and unsteady 2D lid-driven cavity test cases from low Reynolds number (Re=50) to high Reynolds number (Re=2000). The predicted pressure results are compared with the initial pressures, which are traditionally used in the Computational Dynamics Fluids (CFD) iterative methods, such as the Jacobi and Gauss-Seidel methods, by their total steps to reach convergence for the Poisson equation of pressure. For the performance of the VAE model, it averagely reduces the steps for convergence for thousands of steps, and approximately 1.2 to 2 times the speedups the traditional CFD methods are reported. This VAE model can be flexibly applied in the pressure projection acceleration as an initial pressure predictor, which is compatible with several iterative methods and allows periodic moving lid and arbitrary length of time-step.

Abstract.................................i
Contents...............................iii
1 Introduction 1
1.1 Objective and motivation 1
1.2 Literature survey 2
1.2.1 Machine Learning for Fluid Dynamics 2
1.2.2 The Fractional Step Method (The Pressure Projection) 2
1.2.3 Supervised Learning for the Prediction of Fluid Dynamics parameters 3
1.2.4 Unsupervised Learning for PDE and Fluid Dynamics 4
1.2.5 AuotoEncoder for Fluid Dynamics 4
1.2.6 Training model – lid driven cavity 5
1.3 Summary 6
2 Methodology 7
2.1 Computer Fluid Dynamics Process 7
2.1.1 Governing Equation 7
2.1.2 Numerical Scheme 8
2.1.3 Complete Solution Procedure 9
2.2 Machine Learning for Pressure Prediction 10
2.2.1 Governing Equation and Pressure Projection 10
2.2.2 Pressure prediction and calculation 11
2.2.3 Variational AutoEncoder (VAE) model 12
3 Numerical result 15
3.1 The database validation 15
3.2 Pressure Reform and Time Reduction 17
3.2.1 Pressure Reform 17
3.2.2 Calculation Time Reduction 18
3.3 Different Loss Functions and Pressure Prediction 24
3.3.1 Error and Loss function 24
3.3.2 Pressure Prediction for 2 loss functions 25
3.4 Untrained Data Prediction 29
3.4.1 Simulation Acceleration of Single Training Data 29
3.4.2 Simulation Acceleration of Hybrid Training Data 30
3.4.3 Simulation Acceleration of the case of higher Re 31
3.4.4 Simulation Acceleration of different time-steps 31
3.4.5 Simulation Acceleration of periodic lid-driven cavity 32
4 Conclusions 37
Bibliography.............................38

[1] Alexandre Joel Chorin. Numerical solution of the navier–stokes equations. Mathematics of Computation, 22(104):745–762, October 1985.
[2] G.K Batchelor. An introduction to fluid dynamics. Cambridge Mathematical Library,
113:1–4, 1967.
[3] Bernd R. Noack Steven L. Brunton and Petros Koumoutsakos. Machine learning for fluid mechanics. Annual Review of Fluid Mechanics, 54:477–508, 2020.
[4] Jessica K. Shang Owen Williams Brian L. Polagye Isabel Scherl, Benjamin Strom and Steven L. Brunton. Robust principal component analysis for modal decomposition of corrupt fluid flows. Physical Review Fluids, 5:054401, 2020.
[5] Mark L. Kimber Paul J. Kristo and Sharath S. Girimaji. Towards reconstruction of complex flow fields using unit flows. Fluids, 6(7):255, 2021.
[6] Luiz S. Martins-Filho Fernando Madeira SungKi Jung, Won Choi. An implementation of self-organizing maps for airfoil design exploration via multi-objective optimization technique. Journal of Aerospace Technology and Management, 5(2):193–202, 2016.
[7] Maziar S. Hemati Hao Zhang Clarence Rowley Eric A. Deem, Louis N. Cattafesta III and Rajat Mitta. Adaptive separation control of a laminar boundary layer using online dynamic mode decomposition. Journal of Fluid Mechanics, 903:A21, 2020.
[8] Christopher J. Landry Maˇsa Prodanovi´c Javier E.Santos, Duo Xu Honggeun Jo and Michael J. Pyrcz. Poreflow-net: A 3d convolutional neural network to predict fluid flow through porous media. Advances in Water Resources, 138:103539, 2020.
[9] Jean Rabault Aur´elien Larcher Alexander Kuhnle Paul Garnier, Jonathan Viquerat and Elie Hachem. A review on deep reinforcement learning for fluid mechanics. Computers and Fluids, 225:104973, 2021.
[10] J. Kim and P. Moin. Application of a fractional step method to incompressible navier–stokes equations. Journal of Computational Physics, 59:308–323, 1985.
[11] Francis Giraldo Wang Chang and Blair Perot. Analysis of an exact fractional step method. Journal of computational physics, 180:183–199, 2002.
[12] Dana Jacobsen and Inanc Senocak. A parallel multilevel preconditioned iterative pressure poisson solver for the large-eddy simulation of turbulent flow inside a duct. 49th AIAA Aerospace Sciences Meeting, 2012.
[13] Dana Jacobsen Rey DeLeon and Inanc Senocak. Large-eddy simulations of turbulent incompressible flows on gpu clusters. Computing in Science and Engineering, 15:26–33, 2013.
[14] Achi Brandt. Multi-level adaptive solutions to boundary-value problems. Mathematics of
Computation, 31:333–390, 1977.
[15] Zih-Hao Wei Sheng-Hong Lai Chao-An Lin Hsin-Wei Hsu, Feng-Nan Hwang. A parallel multilevel preconditioned iterative pressure poisson solver for the large-eddy simulation of turbulent flow inside a duct. Computers and Fluids, 45:138–146, 2011.
[16] Barbara Solenthalery Marc Pollefeys L’ubor Ladicky, SoHyeon Jeong and Markus Gross.
Data-driven fluid simulations using regression forests. ACM Trans. Graph., 199(6):1–9, 2015.
[17] Xubo Yang Cheng Yang and Xiangyun Xiao. Data-driven projection method in fluid simulation. Computer Animation and Virtual Worlds, 27(3-4):415–424, 2016.
[18] R. K. Jaiman T. P. Miyanawalaa. An efficient deep learning technique for the navier-stokes equations: application to unsteady wake flow dynamics. arXiv, Aug 2018.
[19] Phaedon-SteliosKoutsourelakis ParisPerdikaris YinhaoZhua, NicholasZabaras. Physicsconstrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. Journal ofComputationalPhysics, 394:56–81, 2019.
[20] Xianzhong Ma Bin Dong Zichao Long, Yiping Lu. Pde-net: Learning pdes from data.
Proceedings of the 35th International Conference on Machine Learning, 80:3208–3216, 2018.
[21] Maziar Raissi. Deep hidden physics models: Deep learning of nonlinear partial differential equations. Journal of Machine Learning Research, 19:1–24, 2018.
[22] Deep Ray Kjetil O. Lye, Siddhartha Mishra. Deep learning observables in computational fluid dynamics. Journal of Computational Physics, 410:109339, 2020.
[23] Zico Kolter Filipe De Avila Belbute-Peres, Thomas Economon. Proceedings of the 37th International Conference on Machine Learning, title =
Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction, year = 2020, volume = 119, pages = 2402-2411, publisher = PMLR.
[24] Zhiping Mao Lu Lu, Xuhui Meng and George Em Karniadakis. Deepxde: A deep
learninglibrary for solving differentialequations. SIAM REVIEW, 63(1):208–228, 2021.
[25] Pablo Sprechmann Jonathan Tompson, Kristofer Schlachter and Ken Perlin. Accelerating eulerian fluid simulationwith convolutional networks. Proceedings of the 34 th International
Conference on Machine Learning, 70:3424–3433, 2017.
[26] Micha¨el Bauerheim Antony Misdariis Benedicte Cuenot Ekhi Ajuria Illarramendi, Antonio Alguacil and Emmanuel Benazera. Towards an hybrid computational strategy based on deep learning for incompressible flows. AIAA AVIATION 2020 FORUM, page 3058, 2020.
[27] Sylvain Laizet Panagiotis Tzirakis Georgios Rizos Ali Girayhan ozbay, Arash Hamzehloo and Bjo¨rn Schuller. Poisson cnn: Convolutional neural networks for the solution of the poisson equation on a cartesian mesh. Data-Centric Engineering, 2, 2021.
[28] Modular learning in neural networks. Proc. Conf. on AI, 1:279–284, 1987.
[29] James L. McClel David E. Rumelhart and the PDP Research Group. Parallel
distributed processing: Explorations in the microstructure of cognition. Learning Internal Representations by Error Propagation, 1:318–362, 1986.
[30] ZHERONG PAN BO REN PINGCHUAN MA, YUNSHENG TIAN and DINESH
MANOCHA. Fluid directed rigid body control using deep reinforcement learning. ACM Transactions on Graphics, 37(4):1–11, 2018.
[31] Nils Thuerey Theodore Kim Markus Gross1 Byungsoo Kim1, Vinicius C. Azevedo1 and Barbara Solenthaler. Deep fluids: A generative network for parameterized fluid simulations. Computer Graphics Forum, 38(2):59–70, 2019.
[32] M. Becher S. Wiewel and N. Thuerey. Latent-space physics: Towards learning the temporal evolution of fluid flow. Computer Graphics Forum, 38(2):71–82, 2019.
[33] Abhinav Vishnu and Aparna Chandramowliswharan. Cfdnet: A deep learning-based accelerator for fluid simulations. ACM, 20, 2020.
[34] Lionel Agostini. Exploration and prediction of fluid dynamical systems using auto-encoder technology. Physics of Fluids, 32(6):67103, 2020.
[35] Noam Koenigstein Dor Bank and Raja Giryes. Autoencoders. arXiv, 2020.
[36] Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. arXiv, 2014.
[37] Shin CT Ghia U, Ghia KN. High-re solutions for incompressible flow using the navier–stokes equations and a multigrid method. Journal of Computational Physics, (48):387–411, January 1982.
[38] Robert R. Hwang Yih-Ferng Peng, Yuo-Hsien Shiau. Transition in a 2-d lid-driven cavity flow. Computers and Fluids, (32):337–352, 2003.
[39] Mazen Saad Charles-Henri Bruneau. The 2d lid-driven cavity problem revisited. Computers and Fluids, 35:326–348, 2006.
[40] Steven Frankel Kameswararao Anupindi, Weichen Lai. Characterization of oscillatory instability in lid driven cavity flows using lattice boltzmann method. Computers and Fluids, 92:7–21, 2014.
[41] Keller H.B. Schreiber R. Driven cavity flows by efficient numerical techniques. Journal of
Computer Physics, 49, 1983.
[42] Haecheon Choi and Parviz Moin. Effects of the computational time step on numerical solutions of turbulent flow. Journal of Computational Physics, 113:1–4, 1994.