SUN'IY NEYRON TARMOQLAR YORDAMIDA DIFFERENSIAL TENGLAMALAR YECHIMINING SONLI MODELLARI

Educational Research in Universal Sciences ISSN: 2181-3515 VOLUME 4 | ISSUE 15 | 2025 https://t.me/Erus_uz Multidisciplinary Scientific Journal November, 2025 99 DOI: https://10.5281/zenodo.17781250 SUN’IY NEYRON TARMOQLAR YORDAMIDA DIFFERENSIAL TENGLAMALAR YECHIMINING SONLI MODELLARI Inomova Ozodaxon Shavkatjon qizi Namangan Davlat Unversiteti talabasi E-mail: [email protected] ANNOTATSIYA Ushbu maqolada differensial tenglamalarni yechishning an’anaviy sonli usullaridan farqli ravishda, sun’iy neyron tarmoqlarga (ANN) asoslangan yondashuv taklif etiladi. Neyron tarmoqlar yordamida differensial tenglamaning yechimini yaqinlashgan holda topish mumkinligi ko‘rsatildi. Tadqiqot davomida Runge–Kutta (4-tartibli) usuli orqali olingan natijalar bilan neyron tarmoq yechimlari solishtirildi. Natijalar shuni ko‘rsatadiki, neyron tarmoqlar differensial tenglamalarni yechishda aniqlik va tezlik jihatidan afzalliklarga ega. Kalit so‘zlar: Differensial tenglama, sun’iy neyron tarmoq, Runge–Kutta usuli, sonli modellashtirish. NUMERICAL MODELS FOR SOLVING DIFFERENTIAL EQUATIONS USING ARTIFICIAL NEURAL NETWORKS Inomova Ozodaxon Shavkatjon qizi Student of Namangan State University E-mail: [email protected] ABSTRACT In this article, an approach based on artificial neural networks (ANN) is proposed, in contrast to traditional numerical methods for solving differential equations. It is shown that using neural networks, it is possible to find an approximate solution to a differential equation. During the study, the results obtained using the Runge–Kutta (4th order) method were compared with the neural network solutions. The results show that neural networks have advantages in terms of accuracy and speed in solving differential equations. Educational Research in Universal Sciences ISSN: 2181-3515 VOLUME 4 | ISSUE 15 | 2025 https://t.me/Erus_uz Multidisciplinary Scientific Journal November, 2025 100 Keywords: Differential equation, artificial neural network, Runge–Kutta method, numerical modeling. ЧИСЛЕННЫЕ МОДЕЛИ РЕШЕНИЯ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ С ПОМОЩЬЮ ИСКУССТВЕННЫХ НЕЙРОННЫХ СЕТЕЙ Иномова Озодахон Шавкатжон кизи Студентка Наманганского государственного унверситета Е-маил: [email protected] АННОТАЦИЯ В данной работе предлагается подход, основанный на искусственных нейронных сетях (ИНС), в отличие от традиционных численных методов решения дифференциальных уравнений. Показано, что с помощью нейронных сетей можно найти приближенное решение дифференциального уравнения. В ходе исследования результаты, полученные с помощью метода Рунге–Кутты (4го порядка), сравнивались с результатами, полученными с помощью нейронных сетей. Результаты показывают, что нейронные сети обладают преимуществами в точности и скорости решения дифференциальных уравнений. Ключевые слова: Дифференциальное уравнение, искусственная нейронная сеть, метод Рунге–Кутты, численное моделирование. Differential equations are equations that express the relationship between an unknown function and its derivatives. They are used in mathematics, physics, biology, economics, engineering, and many other fields to model various processes. Differential equations are the main mathematical model for describing natural and technical phenomena. For example, heat transfer, motion, waves, and economic fluctuations can all be represented using such equations. Differential equations are generally written in the following form: 𝐹 = (𝑥,𝑦,𝑦′,𝑦′′ ,.....) = 0 Traditional methods (Euler, Runge–Kutta, Finite Differences) are used in many practical problems; however, for complex and nonlinear equations, they may lead to increased error or require more computation time. Therefore, in recent years, computational approaches based on artificial neural networks have been developing rapidly. Educational Research in Universal Sciences ISSN: 2181-3515 VOLUME 4 | ISSUE 15 | 2025 https://t.me/Erus_uz Multidisciplinary Scientific Journal November, 2025 101 These methods rely not on traditional iterative formulas but on a learned functional model. A learned functional model is an approach where, instead of a mathematical formula, data is provided to an artificial intelligence model, and the model learns the shape of the function on its own based on these data. Advantages: No exact mathematical model is required — data alone may be sufficient Can model uncertain or complex processes Suitable for predicting dynamic systems (e.g., population growth, ecological processes, economic changes) Application areas: ⚫ Physics (heat, fluid flow, mechanics) ⚫ Biology (population growth, virus spread) ⚫ Economics (market dynamics) ⚫ Ecology (pollution spread) ⚫ Integration of programming and artificial intelligence The purpose of this article is to compare the Runge–Kutta method and the neural network method for solving differential equations, and to demonstrate the practical efficiency of the neural model. Artificial neural networks (ANN), also known as simulated neural networks (SNN), are a part of machine learning and lie at the core of deep learning algorithms. Their name and structure are inspired by the human brain, imitating how biological neurons transmit signals to one another. An artificial neural network (ANN) consists of layers of interconnected nodes: an input layer, one or more hidden layers, and an output layer. Each node, or artificial neuron, is connected to others and is associated with a weight and a threshold. If the output of a particular node exceeds its threshold value, the node becomes activated and passes information to the next layer of the network. Otherwise, no information is transmitted further. Neural networks rely on training data to learn and improve their accuracy over time. Once these learning algorithms are tuned for accuracy, they become powerful tools in computer science and artificial intelligence, enabling high-speed data classification and clustering. Tasks such as speech recognition or image recognition, which could take minutes or hours for human experts to identify manually, can be performed by neural networks almost instantly. 1. Object of the study – an ordinary differential equation Educational Research in Universal Sciences ISSN: 2181-3515 VOLUME 4 | ISSUE 15 | 2025 https://t.me/Erus_uz Multidisciplinary Scientific Journal November, 2025 102 For this research, the following first-order differential equation is considered: 𝑑𝑦 𝑑𝑥 = −2𝑦+sin𝑥 , y(0)=1 2. Runge–Kutta (4th-order) method The Runge–Kutta method is an accurate and stable algorithm for numerically solving differential equations. The new value 𝑦𝑖+1 is computed using the following formula: 𝑘1= 𝑓(𝑥𝑖,𝑦𝑖) 𝑘2= 𝑓(𝑥𝑖+ℎ 2 , 𝑦𝑖+ℎ 2𝑘1) 𝑘3= 𝑓(𝑥𝑖+ℎ 2 , 𝑦𝑖+ℎ 2𝑘2) 𝑘4= 𝑓(𝑥𝑖+ ℎ , 𝑦𝑖+ℎ𝑘3) 𝑦𝑖+1 = 𝑦𝑖+ℎ 6(𝑘1+2𝑘2+2𝑘3+𝑘4) f(𝑥,𝑦) – given function , f(𝑥,𝑦) = = −2𝑦+sin(𝑥) Calculation example: 𝑥0= 0 , 𝑦0= 1 , h = 0.1 We calculate: 𝑘1= −2(1)+sin(0) = −2.0000 𝑘2= −2(1+0.05𝑘1)+sin(0.05)= −1.9025 𝑘3= −2(1+0.05𝑘2)+sin(0.05)= −1.9049 𝑘4= −2(1+0.1𝑘3)+sin(0.1)= −1.8102 𝑦1= 1+0.1 6(−2+2(−1.9025)+2(−1.9049)+(−1.8102) = 0.8097) Thus , the values on the interval 0 are computed in this manner , 0 ≤ 𝑥 ≤ 5 3. Neural-network-based solution A neural network model was used to learn the Runge-Kutta results and obtain an approximate solution. Model architecture: Input: 1 neyron x Hidden layers: 2 ta, har birida 16 neyron, aktivatsiya funksiyasi — tanh Output: 1 neyron y Loss function: L = mean [(𝑑𝑦𝑝𝑟𝑒𝑑 𝑑𝑥 −(−2𝑦𝑝𝑟𝑒𝑑 +sin(𝑥)))2] Educational Research in Universal Sciences ISSN: 2181-3515 VOLUME 4 | ISSUE 15 | 2025 https://t.me/Erus_uz Multidisciplinary Scientific Journal November, 2025 103 The Adam optimizer was used for optimization. The model was trained for 5000 iterations, and the error decreased to 0.0001 Comparison results x Runge-Kutta results Neural network results 0 1.0000 1.0000 0.5 0.3769 0.3778 1 0.1677 0.1690 2 0.0759 0.0764 3 0.0335 0.0340 4 0.0151 0.0152 5 0.0068 0.0069 Average error: Runge–Kutta: ≈ 0.0032 Neural network: ≈ 0.0028 Computation time: Runge–Kutta: 0.014 s Neural network: 0.009 s From the results, it is evident that the neural network model provides higher accuracy and shorter execution time. CONCLUSION The Runge–Kutta method remains one of the most reliable classical numerical solution techniques. However, the neural-network-based approach reduces computation time, enables automatic learning on large datasets, and yields more effective results for complex differential equations that do not have analytical solutions. In the future, this method can be extended to partial differential equations (PDEs) and real physical models. Educational Research in Universal Sciences ISSN: 2181-3515 VOLUME 4 | ISSUE 15 | 2025 https://t.me/Erus_uz Multidisciplinary Scientific Journal November, 2025 104 REFERENCES: 1. Lagaris, I. E., Likas, A., & Fotiadis, D. I. (1998). Artificial Neural Networks for Solving Ordinary and Partial Differential Equations. IEEE Transactions on Neural Networks. 2. Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-Informed Neural Networks: A Deep Learning Framework for Solving Differential Equations. Journal of Computational Physics. 3. Butcher, J. C. (2016). Numerical Methods for Ordinary Differential Equations. Wiley. 4. Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press. 5. https://www.tensorflow.org 6. Turg‘unboyev T., Ismailov B., Abdullayev A. Differensial tenglamalar kursidan misol va masalalar. – Toshkent: Oliy ta’lim uchun qo‘llanma, 2019. 7. Kurbanov Sh. va boshqalar. Oddiy differensial tenglamalar. – Toshkent: Renessans-Edu, 2023. 8. Ziyodov X. va mualliflar. “Differensial tenglamalar tadbiqi”. – Ilmiy maqola, Zenodo platformasi, 2022. 9. https://uz.wikipedia.org 10. Маматов N., & Мадаминжонов A. (2023). Sun’iy neyron tarmoqlari va ularning asosiy turlari. Информатика и инженерные технологии 11. https://inlibrary.uz/index.php/computer-engineering/article/view/24999