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

• 批准号: 2343135
• 负责人: Yulong Lu
• 金额: $ 20.5万
• 依托单位: University of Minnesota-Twin Cities
• 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: 2023-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2343135&HistoricalAwards=false
• 关键词: Theoretical; Guarantees; Machine; Learning; Methods

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的日常生活中的各种应用（例如面部识别和自动驾驶）提供实用答案方面取得了巨大的经验成功。该项目将开发用于科学和工程中应用问题的机器学习方法的理论基础。这项研究将在确定预测能力并量化应用程序中机器学习方法的鲁棒性和稳定性方面发挥主要作用。研究人员将指导毕业生和本科生，以研究该项目的理论和应用方面。研究人员将在研究生和高级本科级别开发新的数学机器学习课程，向高中生提供宣传课程。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方法，用于解决偏差方程（PDES），强调其高度衡量的性能和不及不确定性的量化。调查人员将研究两个项目。首先是在神经网络解决方案的概括误差和训练动力学上得出新的维度分解融合估计。这取决于在与神经网络相关的新的基于复杂性的功能空间中建立新的规律性理论。第二个目标是量化贝叶斯框架中神经网络预测的不确定性。这项研究将着重研究贝叶斯神经网络的频繁性能和可扩展的后验计算，以解决高维PDES。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估标准来评估的。