TY - JOUR T1 - Orthogonal Embedding-Based Artificial Neural Network Solutions to Ordinary Differential Equations TT - Adi Diferansiyel Denklemlerin Ortogonal Gömme Tabanlı Yapay Sinir Ağı Çözümleri AU - Uçar, Tolga Recep AU - Tali, Hasan Halit PY - 2025 DA - June Y2 - 2025 DO - 10.35414/akufemubid.1558289 JF - Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi PB - Afyon Kocatepe University WT - DergiPark SN - 2149-3367 SP - 489 EP - 496 VL - 25 IS - 3 LA - en AB - Providing numerical solutions to differential equations in cases where analytical solutions are not available is of great importance. Recently, obtaining more accurate numerical solutions with artificial neural network-based machine learning methods are seen as promising developments for numerical solutions of differential equations. In this paper, a low-cost, orthogonal embedding-based network with fast training by simple gradient descent algorithm is proposed to obtain numerical solutions of differential equations. This architecture is essentially a two-layer neural network that takes orthogonal polynomials as input. The efficiency and accuracy of the method used in this paper are demonstrated in various problems and comparisons are made with other methods. It is observed that the proposed method stands out especially when compared with high-cost solutions. KW - non-linear ordinary differential equations KW - numerical approximation KW - artificial neural networks KW - orthogonal polynomials N2 - Analitik çözümlerin mevcut olmadığı durumlarda diferansiyel denklemler için nümerik çözümler elde etmek büyük önem taşımaktadır. Son zamanlarda, yapay sinir ağı tabanlı makine öğrenmesi yöntemleriyle daha tutarlı nümerik çözümlerin elde edilmesi diferansiyel denklemlerin nümerik çözümleri için ümit verici gelişmeler olarak görülmektedir. Bu makalede, diferansiyel denklemlerin nümerik çözümlerini elde etmek için basit gradyan düşüm algoritması ile hızlı eğime sahip düşük maliyetli bir ortogonal gömme tabanlı ağ önerilmektedir. Bu mimari, temelde, ortogonal polinomları girdi olarak alan iki katmanlı bir sinir ağıdır. Bu makalede kullanılan yöntemin verimliliği ve tutarlılığı, çeşitli problemlerde gösterilmiş ve diğer yöntemlerle karşılaştırmalar yapılmıştır. Kullanılan yöntemin, özellikle yüksek maliyetli çözümlerle karşılaştırıldığında öne çıktığı görülmüştür. CR - Chakraverty, S. and Mall, S., 2017. Artificial Neural Networks for Engineers and Scientists. CRC Press. https://doi.org/10.1201/9781315155265 CR - Cybenko, G., 1989. Approximation by superpositions of a sigmoidal function. Math. 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PLOS ONE, 17(4). https://doi.org/10.1371/journal.pone.0265992 UR - https://doi.org/10.35414/akufemubid.1558289 L1 - https://dergipark.org.tr/en/download/article-file/4250487 ER -