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

机器学习和神经网络（NN）对我们日常生活的影响，是其他数学工具从未有过的。机器学习也吸引了科学计算界的狂热兴趣，神经网络技术已经应用于科学计算中的各种应用，包括微分方程的数值解（PDEs）。然而，当涉及到将神经网络应用于科学应用时，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值，由坚实的理论基础支持，给出有意义的结果，可重复，分析人员可理解，并且（理想情况下）具有可靠的精度数值界限。数值模拟通常用于科学和工程应用，以形成关键决策的基础，这些决策需要对数值结果具有一定的最小置信度。机器学习和神经网络技术将不会被接受，除非开发出解决这些最低要求的方法。目前用于科学应用的机器学习和神经网络方法，特别是偏微分方程的数值解，还没有达到这个水平。该项目包括对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，其总体目标是开发有效的计算工具，并由严格的理论支持，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。当前项目的一个关键特征是重视为该方法发展严格的理论基础，包括对方法的收敛性和准确性的先验结果。此外，该提案旨在通过提供可计算的误差后验估计量，为所得近似值的准确性和保真度提供定量估计。本项目开发的技术将适用于在广泛学科范围内产生的广泛类别的pde。即将开发的工具将立即应用于这些领域。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，其影响将超出项目重点的偏微分方程的数值近似。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.