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

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）提供实际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应用中机器学习方法的鲁棒性和稳定性方面发挥主要作用。研究者将指导研究生和本科生在项目的理论和应用方面工作。该研究员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，用于分析求解偏微分方程（PDEs）的基于神经网络的方法，强调其高维性能和不确定性量化。调查员将从事两个项目。首先是在神经网络解的泛化误差和训练动力学上推导新的维显式收敛估计。这依赖于在与神经网络相关的新的基于复杂性的函数空间中为偏微分方程建立新的正则性理论。第二个目标是在贝叶斯框架下量化神经网络预测中的不确定性。本研究将重点研究贝叶斯神经网络在求解高维偏微分方程中的频率性能和可扩展后验计算。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 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.