Neural Network Approximation of PDEs - Efficiency, Reliability and Quantifiable Accuracy

偏微分方程的神经网络逼近 - 效率、可靠性和可量化的准确性

• 批准号: 2324364
• 负责人: Mark Ainsworth
• 金额: $ 49.46万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-10-01 至 2026-09-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2324364&HistoricalAwards=false
• 关键词: Neural; Network; Approximation; PDEs; Efficiency

Machine learning and neural networks (NN) are impacting everyday our lives in a way that few, if any, other mathematical tool has ever done before. Machine learning is also attracting fevered interest in the scientific computing community where neural network techniques have been applied to all manner of applications in scientific computing including the numerical solution of differential equations (PDEs). However, the bar is set much higher when it comes to applying neural networks to scientific applications, where there is an expectation that numerical methods are capable of producing high accuracy approximations, supported by a solid theoretical foundation, give results that make sense, are reproducible, intelligible to the analyst, and (ideally) come with reliable numerical bounds on the accuracy. Numerical simulation is routinely used in science and engineering applications to form the basis for a critical decision for which certain minimal levels of confidence in the numerical results are required. Machine learning and neural network techniques will not be accepted until and unless methods are developed that address these minimal requirements. Current machine learning and neural network methods for scientific applications in general, and numerical solution of PDEs in particular, are not at this level. The project includes graduate student training on the fundamental research questions relating to the project and implementation of numerical algorithms that verify the theoretical analysis.This project builds on foundations in machine learning and neural network approximation with the overall objective of developing effective computational tools supported by rigorous theory that provide the analyst with the confidence to apply neural network based techniques to scientific applications. A key feature of the current project is the importance attached to developing rigorous theoretical foundations for the approach including a priori results on the convergence and accuracy of the methods. In addition, the proposal aims to provide quantitative estimates for the accuracy and fidelity of the resulting approximations through the provision of computable a posteriori estimators for the error. The techniques developed in this project will be applicable to a broad class of PDEs arising across a wide range of disciplines. The tools that will be developed will find immediate application in these areas. An important feature of the project is the development of effective training algorithms for neural networks that will have impact beyond the numerical approximation of PDEs that are the focus of the project.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.

机器学习和神经网络(NN)正在以一种其他数学工具从未有过的方式影响着我们的日常生活。机器学习也引起了科学计算界的浓厚兴趣，神经网络技术已经被应用到科学计算的各种应用中，包括微分方程组的数值求解。然而，将神经网络应用于科学应用的门槛要高得多，在科学应用中，人们期望数值方法能够产生高精度的近似，有坚实的理论基础支持，给出有意义的结果，可重现，分析师可理解，(理想情况下)具有可靠的精度数值界限。在科学和工程应用中，通常使用数值模拟来形成关键决策的基础，而关键决策需要对数值结果进行一定程度的置信度。除非开发出满足这些最低要求的方法，否则机器学习和神经网络技术将不会被接受。目前用于科学应用的机器学习和神经网络方法，特别是偏微分方程组的数值解，还没有达到这个水平。该项目包括与项目相关的基础研究问题的研究生培训，以及验证理论分析的数值算法的实施。该项目建立在机器学习和神经网络近似的基础上，总体目标是开发由严谨理论支持的有效计算工具，为分析师提供将基于神经网络的技术应用于科学应用的信心。当前项目的一个主要特点是重视为该方法发展严格的理论基础，包括关于方法的汇聚性和准确性的先验结果。此外，该提案旨在通过提供误差的可计算的后验估计器，对由此产生的近似值的准确性和保真度提供定量估计。本项目中开发的技术将适用于出现在广泛学科范围内的广泛类别的PDE。将开发的工具将立即在这些领域得到应用。该项目的一个重要特征是为神经网络开发有效的训练算法，其影响将超出作为项目重点的偏微分方程的数值近似。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。