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研究生：	鄭培辰 Cheng, Pei-Chen
論文名稱：	使用深度學習和隨機微分方程式探討材料之等效熱傳導反應 Effective thermal conduction responses of materials analyzed by stochastic differential equations with deep learning
指導教授：	王雲哲 Wang, Yun-Che
學位類別：	碩士 Master
系所名稱：	工學院 - 土木工程學系 Department of Civil Engineering
論文出版年：	2020
畢業學年度：	108
語文別：	英文
論文頁數：	102
中文關鍵詞：	深度神經網路、深度倒向隨機偏微分方程、有限元素方法、等效熱傳導係數、拋物線型偏微分方程
外文關鍵詞：	Deep neural network, Deep backward stochastic differential equations, Finite element method, Effective thermal conductivity, Parabolic-type partial differential equations
相關次數：	點閱：127 下載：0
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拋物線型偏微分方程，例如：熱傳導問題，可以經由應用費曼－卡茨公式（Feynman-Kac formula）或倒向隨機偏微分方程（Backward stochastic differential equation）進行求解。將倒向隨機偏微分方程與深度神經網路進行配合而創造出的DBSDE演算法，可破除計算高維度問題時會面臨的一大問題，維度詛咒（Curse of dimensionality），能有效地對無限域高維度拋物線型方程進行求解。在本研究中，將探討上述方法論並進行展示。此外，在以有限元素法分析兩相複合材料的有效熱傳導係數並將該複合材料均質化後，以DBSDE演算法對其熱擴散行為進行探討，隨後以有限元素法進行結果上的驗證。除熱擴散問題的探討外，也將探討非線性拋物線型偏微分方程，Allen-Cahn方程式以學習在固體材料上發生的相變換。就結論而言，在三維熱傳導問題上，利用DBSDE演算法計算的結果與利用有限元素法計算的結果相比可觀察到最大的相對誤差為4.7%，至於100維相變換問題，在時間區間0到0.3秒中的0到0.1秒，可觀察到相對誤差小於20%，並在u趨近於零後，誤差開始隨時間增長，最後達到誤差100%。最後，透過不同維度的計算探討DBSDE演算法在計算時間比較以及損失函數的收斂情形，可以得知此演算法在計算特定維度區間的問題時，是個很好的選擇，以單一中央處理器的電腦而言，對50維至400維的問題，以給予良好的初始猜測值為前提，所需花費的計算時間是落在合理可接受的範圍內，大約兩分鐘至五分鐘。將此演算法改良至能處理有限域問題以及可直接利用DBSDE求得複合材料的等效熱傳導係數是未來研究的重要方向。

Parabolic-type partial differential equations (PDE), such as the one for the heat conduction problem, can be solved by stochastic methods, such as the Feynman-Kac method, or through backward stochastic differential equations (BSDE). Combining BSDE and deep neural networks (DNN’s), which apply a central role in the development of artificial intelligence, one create the deep BSDE (DBSDE) algorithm, which can break the curse of dimensionality to efficiently solve high-dimensional parabolic-type PDE in infinite domain. In this work, the methodologies of BSDE and DNN are examined. The effective thermal conductivity of two-phase composite materials is analyzed via the finite element method (FEM). Subsequently, the homogenized composite materials are calculated with the DBSDE method to study their heat diffusion behavior. The DBSDE results are compared with the FEM data for validation. In addition to the heat problem, the Allen-Cahn nonlinear parabolic PDE is also analyzed for studying solid phase transition. For the results in two cases of 3-dimensional heat equation, the maximum L1 relative error that can be observed is about 4.7%, as for the results of 100-dimensional Allen-Cahn equation calculated in time interval [0; 0:3], from t = 0 to t = 0:1, error is lower than 20%, after u converged, error start to grow higher with time. By comparing runtime and loss convergence, one can observed that DBSDE is a great option to choose when dimension of the equation to be solved is high but lower than a particular number depending on the performance of hardware. In the case of computing on a single CPU computer, a region from 50 to 400 is suggested for acceptable computation time. As for the future works, algorithms for DBSDE to solve finite-domain PDE’s are needed. Homogenization by directly using DBSDE requires further investigations.

CHINESE ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Goals and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Normalization on diffusion coefficient . . . . . . . . . . . . . . . . . . . . . . 5
3 Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1 Summary of equations . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Constitutive law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Coupled thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Uncoupled thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . 14
4 Stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1 Markov chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Gaussian process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2.1 Proof of expectation of Gaussian process E[X] =  . . . . . 15
4.2.2 Proof of variance of Gaussian process V ar(X) = 2 . . . . . 16
4.3 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 Monte Carlo stochastic simulation . . . . . . . . . . . . . . . . . . . . . . . . 19
5.1 Monte Carlo method and Monte Carlo integration . . . . . . . . . . . . 19
5.2 Monte Carlo integration . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.3 Solving linear ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.4 Solving nonlinear ODEs . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.5 Solving Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6 Branching process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
7 Stochastic differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . 26
7.1 The Feynman-Kac formula . . . . . . . . . . . . . . . . . . . . . . . . 26
7.2 Ordinary forward stochastic differential equations . . . . . . . . . . . . 27
7.3 Backward stochastic differential equation . . . . . . . . . . . . . . . . 27
7.4 Connection to deterministic PDE . . . . . . . . . . . . . . . . . . . . 28
7.5 For n-dimensional parabolic equation . . . . . . . . . . . . . . . . . . 29
8 Initial vs. terminal condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
9 Deep learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Numerical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1 Effective thermal conductivity of spheres in a block . . . . . . . . . . . . . . . 36
2 Solving parabolic PDE through deep learning and BSDE . . . . . . . . . . . . 39
2.1 Discretized equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Neural network construction . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 3-dimensional Heat transfer problem . . . . . . . . . . . . . . . . . . . . . . . 41
3.1 Finite domain vs. Infinite domain . . . . . . . . . . . . . . . . . . . . 41
3.2 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Allen-Cahn equaiton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1 Runtime vs. Step and Step vs. Loss convergence . . . . . . . . . . . . 44
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1 Spheres in a block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2 Comparison of different boundary conditions . . . . . . . . . . . . . . . . . . 54
3 Results of 3-dimensional heat transfer problem . . . . . . . . . . . . . . . . . 59
4 100-dimensional Allen-Cahn equation . . . . . . . . . . . . . . . . . . . . . . 62
5 Performance and efficiency of DBSDE . . . . . . . . . . . . . . . . . . . . . . 64
Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
APPENDICES
Appendix A: Python code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Appendix B: Presentation slides . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
                                    

[1] J. Han, A. Jentzen, and E.Weinan. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505–8510, 2018.
[2] H. B. Demuth, M. H. Beale, O. De J., and M. T. Hagan. Neural network design. Martin Hagan, 2014.
[3] I. Goodfellow, Y. Bengio, and A. Courville. Deep learning. MIT Press, 2016.
[4] M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. F. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Q. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Man´e, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Vi´egas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Q. Zheng. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015.
[5] D. P. Kingma and J. Ba. ADAM: A method for stochastic optimization. arXiv:1412.6980, 2014.
[6] Monte Carlo methods for solving ordinary differential equations. https://jotterbach.github.io/content/posts/mc_ode/2018-08-08-MonteCarloODE/. Accessed: 2019.
[7] Teaching material. Lecture 4: Monte Carlo Integration. https://inst.eecs.berkeley.edu/~cs294-13/fa09/lectures/scribe-lecture4.pdf. Accessed: 2019.
[8] M. N. Akhtar, M. H. Durad, and A. Ahmed. Solving initial value ordinary differential equations by Monte Carlo method. Proc. of IAM, 4(2):149–174, 2015.
[9] S. S. Sawilowsky. You think you’ve got trivials? Journal of Modern Applied Statistical Methods, 2(1):21, 2003.
[10] G. J. Lord, C. E. Powell, and T. Shardlow. An introduction to computational stochastic PDEs, volume 50. Cambridge University Press, 2014.
[11] Teaching material. Lecture 12: stochastic differential equations, diffusion processes, and the Feynman-Kac formula. http://www.stat.uchicago.edu/~lalley/Courses/391/Lecture12.pdf. Accessed: 2019.
[12] H. Pham. Feynman-Kac representation of fully nonlinear PDEs and applications. Acta Mathematica Vietnamica, 40(2):255–269, 2015.
[13] E. Pardoux and S. J. Tang. Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probability Theory and Related Fields, 114(2):123–150, 1999.
[14] H. P. McKean. Application of Brownian Motion to the Equation of Kolmogorov-Petrovskii-Piskunov. Communications on pure and applied mathematics, 28(3):323–331, 1975.
[15] P. Henry-Labordere. Counterparty risk valuation: A marked branching diffusion approach. SSRN 1995503, 2012.
[16] P. Henry-Labordere, X. L. Tan, and N. Touzi. A numerical algorithm for a class of BSDEs via the branching process. Stochastic Processes and their Applications, 124(2):1112–1140, 2014.
[17] M. Baymani, A. Kerayechian, and S. Effati. Artificial neural networks approach for solving Stokes problem. Applied Mathematics, 1(4):288, 2010.
[18] Q. Chan-Wai-Nam, J. Mikael, and X. Warin. Machine learning for semi linear PDEs. Journal of Scientific Computing, 79(3):1667–1712, 2019.
[19] I. E. Lagaris, A. Likas, and D. I. Fotiadis. Artificial neural networks for solving ordinary and partial differential equations. IEEE transactions on neural networks, 9(5):987–1000, 1998.
[20] L. Lu, X. H. Meng, Z. P. Mao, and G. E. Karniadakis. DEEPXDE: A deep learning library for solving differential equations. arXiv:1907.04502, 2019.
[21] B. A. Le, J. Yvonnet, and Q-C. He. Computational homogenization of nonlinear elastic materials using neural networks. International Journal for Numerical Methods in Engineering, 104(12):1061–1084, 2015.
[22] E. Weinan, J. Han, and A. Jentzen. Deep learning-based numerical methods for highdimensional parabolic partial differential equations and backward stochastic differential equations. Communications in Mathematics and Statistics, 5(4):349–380, 2017.
[23] C. Beck, E. Weinan, and A. Jentzen. Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. Journal of Nonlinear Science, 29(4):1563–1619, 2019.
[24] A. Lejay and H. Mardones Gonz´alez. A forward-backward probabilistic algorithm for the incompressible Navier-Stokes equations. hal-02377108v2, 2019.
[25] M. N. O. Sadiku, C. M. Akujuobi, S. M. Musa, and S. R. Nelutary. Direct Monte Carlo simulation of time-dependent problems. The Technology Interface, 8(1), 2007.

校內：2025-07-15公開
校外：不公開電子論文尚未授權公開，紙本請查館藏目錄

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