Physics Informed Neural Network Method For the Numerical Solution of Fractional Diffusion Equations

Öz

Artificial neural networks are increasingly used to construct continuous solution functions for solving various kinds of differential equations. In this study, we propose a physics informed neural network (PINN) method to solve fractional diffusion equations with variable coefficients on a finite domain. The PINN generate approximate solutions to the fractional PDE by training to minimize the physical loss function consisting of residual, boundary condition and initial condition parts. Fractional PDE is discretized with the Grunwald-Letnikov formula and the resulted semi-discrete equation is used to construct the residual function of the PINN. Numerical experiments show that the present PINN method provides accurate solutions on the considered computational space-time domain.

Anahtar Kelimeler

• Fractional Diffusion Equations
• PINN
• Grünwald-Letnikov formula

Kesirli Difüzyon Denklemlerinin PINN Metodu ile Sayısal Çözümleri

Öz

Yapay sinir ağları, çeşitli diferansiyel denklemlerin çözümünde sürekli çözüm fonksiyonları oluşturmak için giderek daha fazla kullanılmaktadır. Bu çalışmada, sonlu bir alan üzerinde değişken katsayılı kesirli difüzyon denklemlerini çözmek için bir fizik tabanlı sinir ağı (PINN) yöntemi öneriyoruz. PINN, kalıntı, sınır koşulu ve başlangıç koşulu parçalarından oluşan hata fonksiyonunu en aza indirmek için eğitilerek kesirli PDE'ye yaklaşık çözümler üretir. Kesirli PDE, Grunwald-Letnikov formülü ile diskritize edilir ve elde edilen yarı diskrit denklem, PINN'nin hata fonksiyonunu oluşturmak için kullanılır. Sayısal örnekler, mevcut PINN yönteminin dikkate alınan hesaplama uzay-zaman alanı üzerinde doğru çözümler sağladığını göstermektedir. Translated with DeepL.com (free version)

Anahtar Kelimeler

• Kesirli Difüzyon Denklemi
• PINN Metodu
• Nümerik Yöntemler

Kaynakça

1. [1] Mall S., Chakraverty S.,(2013) Comparison of artificial neural network architecture in solving ordinary differential equations, Adv. Artif. Neural Syst., 2013 , p. 12
2. [2] Sabir Z., Baleanu D., Shoaib M., Raja M.A.Z., (2021) Design of stochastic numerical solver for the solution of singular three-point second-order boundary value problems, Neural Comput. Appl., 33 (7) , pp. 2427-2443.
3. [3] Raissi M., Perdikaris P., Karniadakis G.E., (2019) Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 , pp. 686-707.
4. [4] Dwivedi V., Parashar N., Srinivasan B., (2021) Distributed learning machines for solving forward and inverse problems in partial differential equations, Neurocomputing, 420 , pp. 299-316.
5. [5] Cai S., Wang Z., Wang S., Perdikaris P., Karniadakis G.E., (2021) Physics-informed neural networks for heat transfer problems, J. Heat Transfer, 143 (6) .
6. [6] Mortari D., (2017) The theory of connections: Connecting points, Mathematics, 5 (4) , p. 57.
7. [7] Mai T., Mortari D., (2022) Theory of functional connections applied to quadratic and nonlinear programming under equality constraints, J. Comput. Appl. Math., 406 .
8. [8] Li, X. (2012) Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method, Commun Nonlinear Sci Numer Simulat 17 3934–3946.

9. [9] Diethelm K., Ford N.J., (2002) Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2) , pp. 229-248.
10. [10] Kilbas A.A., Srivastava H.M., Trujillo J.J., (2006) Theory and Applications of Fractional Differential Equations, Vol. 204 Elsevier
11. [11] Podlubny I., (1998) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications Elsevier
12. [12] Caputo M., Fabrizio M. A new definition of fractional derivative without singular kernel Prog. Fract. Differ. Appl., 1 (2) , pp. 73-85
13. [13] Atangana A., Baleanu D., (2015) New fractional derivatives with non-local and non-singular kernel, theory and application to heat transfer model, Therm. Sci., 20 (2) (2016), Article 763769.
14. [14] Jafarian A., Mokhtarpour M., Baleanu D., (2017) Artificial neural network approach for a class of fractional ordinary differential equation, Neural Comput. Appl., 28 (4) , pp. 765-773.
15. [15] Sivalingam S.M., Kumar P., Govindaraj V., (2023) The hybrid average subtraction and standard deviation based optimizer, Adv. Eng. Softw., 176 , Article 103387.
16. [16] Alkan, A. (2022). Improving homotopy analysis method with an optimal parameter for time- fractional Burgers equation. Karamano˘glu Mehmetbey ¨Universitesi M¨uhendislik ve Do˘ga Bilimleri Dergisi, 4(2), 117-134.
17. [17] Alkan, A. (2022). Improving Homotopy Analysis Method with An Optimal Parameter for Time- Fractional Burgers Equation. Karamano˘glu Mehmetbey ¨Universitesi M¨uhendislik Ve Do˘ga Bilimleri Dergisi, 4(2), 117-134. https://doi.org/10.55213/kmujens.1206517
18. [18] Alkan, A., & Ana¸c, H. (2024). The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method. AIMS Mathematics, 9(9), 25333-25359.
19. [19] Alkan, A., & Ana¸c, H. (2024). A new study on the Newell-Whitehead-Segel equation with Caputo- Fabrizio fractional derivative. AIMS Mathematics, 9(10), 27979-27997.
20. [20] Bekta¸s, U., & Ana¸c, H. (2024). A hybrid method to solve a fractional-order Newell–Whitehead–Segel equation. Boundary Value Problems, 2024(1), 38.
21. [21] Pakdaman M., Ahmadian A., Effati S., Salahshour S., Baleanu D., (2017) Solving differential equa- tions of fractional order using an optimization technique based on training artificial neural network, Appl. Math. Comput., 293 , pp. 81-95.
22. [22] Pang G., Lu L., Karniadakis G.E., (2019) fPINNs: Fractional physics-informed neural networks, SIAM J. Sci. Comput., 41 (4) , pp. A2603-A2626.
23. [23] Wang, W., Wang K., Sircar, T. (2019) A direct 0(N log2N ) finite difference method for fractional diffusion equations, Journal of Computational Physics, 229 8095-8104.
24. [24] Caglar, S., Ucar, M.F., Caglar, N., Akkoyunlu, C., (2012) Non Polynomial Spline Method for Frac- tional Diffusion Equation, Journal of Computational Analysis and Applications, 1354 - 1361.
25. [25] Tadjeran, C., Meerschaert, M. M., (2007) A second-order accurate numerical method for the two- dimensional fractional diffusion equation, Journal of Computational Physics, 220 813–823