应用与计算数学杂志

Machine Learning-driven Numerical Solutions to Partial Differential Equations

Abstract

Scott Rolfo*

Partial differential equations are fundamental mathematical tools used to describe a wide range of physical phenomena, from fluid dynamics and heat conduction to quantum mechanics and financial modeling. Solving PDEs is crucial for understanding and predicting the behavior of these systems, but traditional numerical methods, such as finite difference, finite element, and spectral methods, often encounter significant challenges when dealing with complex, high-dimensional problems. In recent years, machine learning has emerged as a powerful alternative or complement to classical numerical methods, offering new approaches for efficiently solving PDEs. Machine learning-driven numerical solutions to PDEs have the potential to revolutionize computational science by providing more accurate, faster, and scalable solutions. One of the key motivations for integrating machine learning with numerical PDE solvers is the ability of ML models to approximate complex functions and their derivatives with high accuracy. Neural networks, particularly deep learning models, have demonstrated remarkable success in learning intricate patterns and relationships within large datasets.