Ghani Irfan
, Haider Zaman, Muhammad Ishaq, Wajid Khan
This paper presents an innovative method for solving systems of ordinary differential equations (ODEs) characterized by oscillatory solutions, utilizing Residual-Based Adaptive Refinement of Physics-Informed Neural Networks (RAR-PINNs). Conventional numerical techniques often face challenges in accurately resolving oscillatory solutions due to issues with convergence and stability. To address these challenges, we introduce a refined approach that integrates adaptive refinement strategies with physics-informed neural networks, enhancing their capability to model and predict complex oscillatory dynamics. Our method involves an adaptive mechanism that selectively refines the neural network focus based on the residual errors of the predicted solutions, thereby improving accuracy where it is most needed. By incorporating physical constraints directly into the learning process, our approach ensures that the neural network not only captures the underlying oscillatory patterns but also adheres to the governing differential equations. We validate the effectiveness of the RAR-PINNs approach through numerical experiments on benchmark problems with known oscillatory solutions, demonstrating substantial improvements in both solution accuracy and computational efficiency compared to traditional methods. This advancement provides a powerful tool for tackling highly oscillatory ODE systems in various scientific and engineering applications where oscillatory behavior is prevalent.
Ghani Irfan Haider Zaman Muhammad Ishaq Wajid Khan “Solving System of Highly Oscillatory Ordinary Differential Equations with Resid Vol. 11 Issue 09 PP. 160-164 September 2024. https://doi.org/10.34259/ijew.24.1109160164.
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