Hamilton sistemlerin yapay sinir ağlarıyla yaklaşık çözümü
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Date
2023
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Publisher
Tekirdağ Namık Kemal Üniversitesi
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info:eu-repo/semantics/openAccess
Abstract
Bu çalışmada basit mekanik sistemlerin zamana bağlı değişimlerini açıklayan adi diferansiyel denklemlerin çözümlerine derin öğrenme algoritmaları ile yaklaşım yapılmıştır. Beş bölümden oluşan tezin birinci bölümünde konunun tarihi gelişimi ifade edilmiş ve çalışmanın amacı açıklanmıştır. İkinci bölümde çalışmada geçen temel kavramlar açıklanmıştır. Üçüncü bölümde diferansiyel denklemlerin yapay sinir ağları ile çözümü incelenmiştir. Dördüncü bölümde Hamilton sistemler ve simplektik dönüşümler açıklanmıştır. Çalışmada basit Hamilton sistemler için simplektik Euler, Störmer-Verlet gibi iteratif bir metotla çözülüp elde edilen veri noktaları arttırılarak ve enerji koruyan kayıp fonksiyonu kullanılarak veri odaklı yapay sinir ağları ile çözüme yaklaşım sağlanmıştır. Beşinci bölümde önerilen yöntem basit harmonik osilatör, lineer olmayan osilatör ve Lotka-Volterra denklemlerine uygulanmış ve iteratif metotlar ile karşılaştırılmıştır.
In this study, ordinary differential equations explaining the time dependent variation of simple mechanical systems are solved using deep learning methods. In the first part of the thesis, which consists of five parts, the historical development of the subject is expressed and the purpose of the study is explained. In the second part, the basic concepts in the study are explained. Then, the solution of differential equations with artificial neural networks is examined. In part 4, Hamilton systems and Symplectic transformations are explained. In the study, an approach to the solution with data driven artificial neural networks is provided by increasing the data points obtained by solving with an iterative method such as Symplectic Euler, Verlet for simple Hamiltonian systems and using the energy-conserving loss function. In the 5. chapter, the proposed method is applied to simple harmonic oscillator, nonlinear oscillator and Lotka-Volterra equation and compared with iterative methods.
In this study, ordinary differential equations explaining the time dependent variation of simple mechanical systems are solved using deep learning methods. In the first part of the thesis, which consists of five parts, the historical development of the subject is expressed and the purpose of the study is explained. In the second part, the basic concepts in the study are explained. Then, the solution of differential equations with artificial neural networks is examined. In part 4, Hamilton systems and Symplectic transformations are explained. In the study, an approach to the solution with data driven artificial neural networks is provided by increasing the data points obtained by solving with an iterative method such as Symplectic Euler, Verlet for simple Hamiltonian systems and using the energy-conserving loss function. In the 5. chapter, the proposed method is applied to simple harmonic oscillator, nonlinear oscillator and Lotka-Volterra equation and compared with iterative methods.
Description
Fen Bilimleri Enstitüsü, Matematik Ana Bilim Dalı
Keywords
Matematik, Mathematics
