CAREER: Gaussian Processes for Scientific Machine Learning: Theoretical Analysis and Computational Algorithms

机器学习和人工智能正在越来越多地改变我们的生活，从我们个人的日常活动到我们社会、经济和地缘政治的大规模转变。这些技术也深刻地改变了科学，新的想法和算法正在以极快的速度发展。然而，我们对这些算法的数学理解仍然远远超出了它们的实际发展和广泛采用。简而言之，在许多情况下，我们不知道机器学习算法如何或为什么工作得如此之好，这反过来限制了我们在安全关键工程和科学场景中安全部署它们的能力。该项目的目标是发展数学形式主义和对机器学习和科学交叉点问题的理解（即，科学机器学习），导致新的算法和计算方法是可解释的，由严格的理论支持，并意识到的不确定性.该项目的重点是开发新的高斯过程（GP）为基础的计算框架的科学机器学习是可证明的适定性，鲁棒性，稳定，从而满足科学计算的高标准.所开发的方法将能够进行严格的不确定性量化，并继承机器学习算法的理想特性，如灵活性，泛化性和高维的适用性。该项目的努力主要集中在三个方向：（1）用于解决非线性，高维和参数偏微分方程的GP;（2）用于算子学习，仿真和物理发现的GP;以及（3）用于高维采样，推理和生成建模的GP。每个研究方向都侧重于算法，基础理论以及工程和科学中的具体应用的发展。该项目还包含一个广泛的教育计划，重点关注从高中到研究生的机器学习和数据科学教育，为研究生和本科生提供广泛的培训机会，以及本地和国际推广。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响力审查标准进行评估，被认为值得支持。

Machine learning and artificial intelligence are increasingly changing our lives at every scale, from our personal day-to-day activities to large scale shifts in our society, economy, and geopolitics. These technologies have also profoundly transformed sciences with new ideas and algorithms being developed at an immense speed. However, our mathematical understanding of these algorithms is still far beyond their practical development and widespread adoption. Put simply, in many cases we do not know how or why machine learning algorithms work so well, which in turn limits our ability to safely deploy them in safety critical engineering and scientific scenarios. The goal of this project is to develop mathematical formalism and understanding of problems at the intersection of machine learning and science (i.e., scientific machine learning) using rigorous mathematics, leading to novel algorithms and computational methodologies that are interpretable, supported by rigorous theory, and aware of uncertainties.The project is focused on the development of novel Gaussian Process (GP) based computational frameworks for scientific machine learning that are provably well-posed, robust, and stable, thereby meeting the high standards of scientific computing. The developed methodologies will be capable of rigorous uncertainty quantification and inherit the desirable properties of machine learning algorithms such as flexibility, generalizability, and applicability in high-dimensions. The efforts of the project are directed in three primary directions: (1) GPs for solving nonlinear, high-dimensional and parametric PDEs; (2) GPs for operator learning, emulation, and physics discovery; and (3) GPs for high-dimensional sampling, inference, and generative modeling. Each research direction focuses on the development of algorithms, foundational theory, and concrete applications in engineering and science. The project also contains an extensive education plan focused on machine learning and data science education from high-school through graduate levels with extensive opportunities for training of graduate and undergraduate students as well as local and international outreach.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.