Survey of AI-Driven approaches for Solving Nonlinear Partial Differential Equations

Nonlinear partial differential equations (PDEs) form the mathematical backbone for modeling phenomena across diverse fields such as physics, biology, engineering, and finance. Traditional numerical methods have limitations, particularly for high- dimensional or parameterized problems, due to the "curse of dimensionality" and computational expense. Artificial Intelligence (AI) is currently a valuable tool and has extensive applications in various fields. AI-driven approaches offer a promising alternative by leveraging machine learning techniques to efficiently approximate solutions, especially in high-dimensional or complex problems. This paper surveys state-of-the-art AI techniques for solving nonlinear PDEs, including Physics-Informed Neural Networks (PINNs), Deep Galerkin Methods (DGM), and Neural Operators. Symbolic computation methods, Hirota bilinear methods, bilinear neural network methods. We explore their theoretical foundations, architectures, advantages, limitations, and applications. Finally, we discuss open challenges and future directions in the field.