Leveraging Artificial Intelligence for the Solution of Differential Equations: A Novel Approach
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Leveraging Artificial Intelligence for the Solution of Differential Equations: A Novel Approach
Kirti Kumar Jain* Sarla Raigar**Harsha Tavse***Manoj Sharma****
*Asst. Professor, Applied Science Department Sagar institute of research and technology Bhopal
**Asst. Professor, Applied Science Department Sagar institute of research and technology Bhopal
***Asst. Professor, Applied Science Department Sagar institute of research and technology Bhopal
**** Professor, Applied Science Department Sagar institute of research and technology Bhopal
Abstract:
Differential equations are fundamental to the modeling of dynamical systems in various scientific fields, yet solving them analytically remains challenging in many cases. This paper explores the application of artificial intelligence (AI) methods, particularly machine learning algorithms, to solve complex differential equations. We propose a new approach using deep learning models such as neural networks to approximate solutions to both ordinary and partial differential equations without the need for closed-form analytical solutions. The models are trained using datasets generated from known solutions, providing flexibility and adaptability to changing boundary conditions and system dynamics. By comparing AI-generated solutions with traditional numerical methods (e.g., finite difference or finite element methods), we demonstrate the potential of AI to reduce computational time, increase accuracy, and handle traditionally intractable high-dimensional problems. This research also investigates the interpretability of AI-based solutions and provides insights into their robustness across a range of scientific and engineering applications. Our results indicate that AI offers a promising alternative to traditional methods for solving differential equations, especially in scenarios with complex, nonlinear dynamics or where data-driven models are preferred.
Keywords:
Artificial Intelligence, Machine Learning, Differential Equations, Neural Networks, Deep Learning, Numerical Methods, Ordinary Differential Equations (ODEs), Partial Differential Equations (PDEs), Computational Modeling, Data-Driven Solutions, Nonlinear Dynamics, Boundary Conditions, Scientific Computing, Approximation Methods, High-Dimensional Problems.
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