论文标题
无监督的Legendre-Galerkin神经网络，用于奇异的偏微分方程
Unsupervised Legendre-Galerkin Neural Network for Singularly Perturbed Partial Differential Equations
论文作者
论文摘要
机器学习方法最近已用于求解部分微分方程（PDE）和动态系统。这些方法已发展为一个新型的研究领域，称为科学机器学习，其中，深层神经网络和统计学习等技术应用于应用数学的经典问题。在本文中，我们开发了一种新颖的数值算法，该算法结合了机器学习和人工智能来解决PDE。基于Legendre-Galerkin框架，我们提出了{\ IT无监督的机器学习}算法，以了解不同类型的PDE的解决方案的{\ IT多个实例}。我们的方法克服了基于数据驱动和物理学的方法的局限性。所提出的神经网络应用于具有各种边界条件的一般1D和2D PDE，以及以对流为主的{\ IT极度扰动的PDES}，表现出强大的边界层行为。
Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques such as deep neural networks and statistical learning are applied to classical problems of applied mathematics. In this paper, we develop a novel numerical algorithm that incorporates machine learning and artificial intelligence to solve PDEs. Based on the Legendre-Galerkin framework, we propose the {\it unsupervised machine learning} algorithm to learn {\it multiple instances} of the solutions for different types of PDEs. Our approach overcomes the limitations of data-driven and physics-based methods. The proposed neural network is applied to general 1D and 2D PDEs with various boundary conditions as well as convection-dominated {\it singularly perturbed PDEs} that exhibit strong boundary layer behavior.
