nonlinear PDEs computational techniques machine learning numerical methods hybrid framework applied mathematics

Nonlinear partial differential equations (PDEs) represent a cornerstone of applied mathematics, with significant implications across physics, engineering, and finance. This paper aims to explore advanced computational methods to solve these complex equations efficiently. We begin by providing a comprehensive review of current analytical and numerical techniques, emphasizing their limitations in handling high-dimensional systems. Our objective is to develop a hybrid computational framework that leverages machine learning algorithms alongside traditional numerical methods. This framework is designed to reduce computation time while maintaining high accuracy. Using a series of benchmark nonlinear PDEs, we implemented our proposed method and compared its performance against conventional approaches. Our findings reveal a marked improvement in speed and precision, particularly in scenarios involving complex boundary conditions. We conclude that our hybrid approach represents a significant advancement in the computational toolkit available to mathematicians and engineers dealing with nonlinear PDEs. Future research will focus on refining the algorithm to handle even more complex systems and exploring its applications in real-world problems.

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