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

Abstract

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.

Keywords

• Fractional Diffusion Equations
• PINN
• Grünwald-Letnikov formula

Kesirli Difüzyon Denklemlerinin PINN Metodu ile Sayısal Çözümleri

Abstract

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)

Keywords

• Kesirli Difüzyon Denklemi
• PINN Metodu
• Nümerik Yöntemler

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