Deep Learning-Based Numerical Solutions for Nonlinear Partial Differential Equations in Fluid Dynamics

Abstract

Nonlinear partial differential equations (PDEs) play a vital role in modeling complex fluid dynamics phenomena such as turbulence, heat transfer, and flow instability. Traditional numerical approaches, including finite difference and finite element methods, often require extensive computational resources and may struggle with large-scale or real-time simulations. Recent advancements in deep learning have introduced efficient alternatives for solving nonlinear PDEs through data-driven and physics-informed approaches. This paper presents a comprehensive study of deep learning-based numerical solutions for nonlinear PDEs in fluid dynamics. The proposed framework integrates neural network architectures with physics informed constraints to accurately approximate fluid behavior while reducing computational complexity. Various deep learning models, including Physics-Informed Neural Networks (PINNs), Deep Operator Networks, and Fourier Neural Operators, are analyzed and compared with conventional numerical techniques. Experimental results demonstrate improved prediction accuracy, faster convergence, and enhanced scalability in solving complex fluid flow problems. The study further discusses practical applications, current limitations, and future research opportunities in AI-driven scientific computing for fluid dynamics.