Comparative Study of Finite Element and Neural Network Discretizations for Partial Differential Equations

偏微分方程有限元与神经网络离散化的比较研究

• 批准号: 2424305
• 负责人: Jonathan Siegel
• 金额: $ 55万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-03-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2424305&HistoricalAwards=false
• 关键词: Comparative; Study; Finite; Element; Neural

This research connects two different fields, machine learning from data science and numerical partial differential equations from scientific and engineering computing, through the comparative study of the finite element method and finite neuron method. Finite element methods have undergone decades of study by mathematicians, scientists and engineers in many fields and there is a rich mathematical theory concerning them. They are widely used in scientific computing and modelling to generate accurate simulations of a wide variety of physical processes, most notably the deformation of materials and fluid mechanics. By contrast, deep neural networks are relatively new and have only been widely used in the last decade. In this short time, they have demonstrated remarkable empirical performance on a wide variety of machine learning tasks, most notably in computer vision and natural language processing. Despite this great empirical success, there is still a very limited mathematical understanding of why and how deep neural networks work so well. We hope to leverage the success of deep learning to improve numerical methods for partial differential equations and to leverage the theoretical understanding of the finite element method to better understand deep learning. The interdisciplinary nature of the research will also provide a good training experience for junior researchers. This project will support 1 graduate student each year of the three year project. Piecewise polynomials represent one of the most important functional classes in approximation theory. In classical approximation theory and numerical methods for partial differential equations, these functional classes are often represented by linear functional spaces associated with a priori given grids, for example, by splines and finite element spaces. In deep learning, function classes are typically represented by a composition of a sequence of linear functions and coordinate-wise non-linearities. One important non-linearity is the rectified linear unit (ReLU) function and its powers (ReLUk). The resulting functional class, ReLUk-DNN, does not form a linear vector space but is rather parameterized non-linearly by a high-dimensional set of parameters. This function class can be used to solve partial differential equations and we call the resulting numerical algorithms the finite neuron method (FNM). Proposed research topics include: error estimates for the finite neuron method, universal construction of conforming finite elements for arbitrarily high order partial differential equations, an investigation into how and why the finite neuron method gives a much better asymptotic error estimate than the corresponding finite element method, and the development and analysis of efficient algorithms for using the finite neuron method.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这项研究通过对有限元方法和有限神经元方法的比较研究，将两个不同的领域从数据科学和数值偏微分方程中连接起来。有限元方法已经在许多领域的数学家，科学家和工程师进行了数十年的研究，并且有关于它们的丰富数学理论。它们被广泛用于科学计算和建模中，以生成各种物理过程的准确模拟，最著名的是材料和流体力学的变形。相比之下，深度神经网络相对较新，并且仅在过去十年中被广泛使用。在此短时间内，他们在各种机器学习任务上表现出了出色的经验表现，最著名的是在计算机视觉和自然语言处理中。尽管取得了巨大的经验成功，但仍然对为什么和深层神经网络的运作良好的数学了解仍然非常有限。我们希望利用深度学习的成功来改善部分微分方程的数值方法，并利用对有限元方法的理论理解来更好地了解深度学习。研究的跨学科性质还将为初级研究人员提供良好的培训经验。该项目将每年三年项目的每年支持1名研究生。分段多项式代表近似理论中最重要的功能类别之一。 在偏微分方程的经典近似理论和数值方法中，这些功能类通常由与先验给定的网格相关的线性功能空间表示，例如，花键和有限元空间。在深度学习中，功能类通常由线性函数序列和坐标的非线性组成表示。一个重要的非线性性是整流线性单元（RELU）函数及其功能（RELUK）。 由此产生的功能类Reluk-DNN不形成线性向量空间，而是通过一组高维参数的参数性来非线性化。 该功能类可用于求解部分微分方程，我们将所得的数值算法称为有限神经元方法（FNM）。拟议的研究主题包括：有限神经元方法的误差估计，对任意高阶部分偏微分方程的有限元素的普遍构建，对有限神经元方法的调查以及为什么对相应的有限元方法的渐近误差估计更好，以及对相应的有限元方法的更好估计，以及对使用有限的神经元的有效algority nears and decrient and necors necors nectors ne sef. thes thes.thes.the.值得通过基金会的智力优点和更广泛的影响审查标准来通过评估来支持。