Affiliation:
1. Government College for Men, Cluster University, B-Camp, Kurnool, Andhra Pradesh, India
2. Government Degree College, Kukatpally, Hyderabad, India
Abstract
The intersection of artificial intelligence (AI) and partial differential equations (PDEs), emphasizing how AI techniques can revolutionize the analysis and solution of PDEs in various scientific and engineering applications. Traditional methods for solving PDEs often face challenges related to computational complexity, high-dimensionality, and nonlinearity. By leveraging advanced AI algorithms, particularly deep learning and neural networks, we propose novel approaches to approximate solutions, reduce computational costs, and handle complex boundary conditions more effectively. The study highlights the advantages of AI-driven methods in terms of accuracy, efficiency, and scalability, presenting case studies from fluid dynamics, quantum mechanics, and financial mathematics. Our findings suggest that AI has the potential to significantly enhance the analytical capabilities and practical applications of PDEs, paving the way for new advancements in both theoretical research and real-world problem solving
Reference21 articles.
1. Raissi, Maziar, Paris Perdikaris, & George EmKarniadakis. “Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations,” Journal of Computational Physics, 2017.
2. J. Sirignano, and Spiliopoulos,”KDGM: A deep learning algorithm for solving partial differential equations,” Journal of Computational Physics,375,1339-1364,2018.
3. N. Geneva, andN. Zabaras, “Modeling the Dynamics of PDE Systems with Physics- Constrained Deep Auto-Regressive Networks,” Journal of Computational Physics, 2019.
4. A. J. Chorin, and M.Morzfeld, “A numerical method for solving incompressible viscous flow problems,” Journal of Computational Physics, 229(23), 9234-9247, 2010.
5. J. Sirignano, and M. Spiegelman, “DarcyTools: A Python package for inversion and spectral analysis of Darcy flow in heterogeneous porous media,” Computational Geosciences,22(5), 1277-1292, 2018.