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Theoretical Guarantees of Machine Learning Methods for High Dimensional Partial Differential Equations: Numerical Analysis and Uncertainty Quantification

高维偏微分方程机器学习方法的理论保证：数值分析和不确定性量化

基本信息

批准号：
2107934
负责人：
Yulong Lu
金额：
$ 20.5万
依托单位：
University of Massachusetts Amherst
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-01 至 2023-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2107934&HistoricalAwards=false
关键词：
Theoretical Guarantees Machine Learning Methods

项目摘要

Machine learning algorithms have achieved tremendous empirical successes in providing practical answers to various applications in our everyday life, such as face recognition and autonomous driving. This project will develop theoretical foundations of machine learning methods for applied problems in science and engineering. The research will play a principal role in determining predictive power and quantifying the robustness and stability of the machine learning methodology in applications. The investigator will mentor graduate and undergraduate students to work on both theoretical and applied aspects of the project. The investigator will provide outreach to high school students with an introductory course on Data Science and develop new mathematical machine learning courses at both graduate and advanced undergraduate levels.The project will develop a systematic mathematical framework for analyzing neural network-based methods for solving partial differential equations (PDEs), emphasizing their high-dimensional performance and uncertainty quantification. The investigator will work on two projects. The first is to derive new dimension-explicit convergence estimates on the generalization error and training dynamics of neural network solutions. This relies on establishing a new regularity theory for PDEs in new complexity-based function spaces tied to neural networks. The second objective is to quantify the uncertainty in the neural network prediction in a Bayesian framework. The research will focus on studying the frequentist performance and the scalable posterior computation of the Bayesian neural networks for solving high dimensional PDEs.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.

机器学习算法在为我们日常生活中的各种应用（如人脸识别和自动驾驶）提供实用答案方面取得了巨大的经验成功。该项目将为科学和工程中的应用问题开发机器学习方法的理论基础。该研究将在确定预测能力和量化机器学习方法在应用中的鲁棒性和稳定性方面发挥主要作用。研究员将指导研究生和本科生在该项目的理论和应用方面开展工作。该项目将为高中生提供数据科学入门课程，并为研究生和高级本科生开发新的数学机器学习课程。该项目将开发一个系统的数学框架，用于分析基于神经网络的偏微分方程（PDE）求解方法，强调其高维性能和不确定性量化。调查员将从事两个项目。第一个是得到新的维数显式收敛估计的泛化误差和训练动态的神经网络解决方案。这依赖于建立一个新的正则性理论的偏微分方程在新的复杂性为基础的功能空间与神经网络。第二个目标是在贝叶斯框架中量化神经网络预测的不确定性。该研究将侧重于研究频率论性能和贝叶斯神经网络解决高维偏微分方程的可扩展后验计算。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。