Efficient Algorithms Related to and Beyond the Large Deviation Technique

与大偏差技术相关及之外的高效算法

• 批准号: 1913163
• 负责人: Xiaoliang Wan
• 金额: $ 19.58万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-01 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1913163&HistoricalAwards=false
• 关键词: Efficient; Algorithms; Related; Beyond; Large

Mathematical modeling methods based on partial differential equations are widely used in engineering and scientific applications, which have been one of the most important tool for mankind to understand a large variety of phenomena originating from human activity and technological development. The stochastic partial differential equations generalize the partial differential equations by taking into account the uncertainty, which is ubiquitous in reality. In this project, we focus on the simulation and quantification of rare events in stochastic partial differential equations that can model some important phenomena such as regime change in climate, rogue ocean waves, abnormal weather, etc, which may occur rarely but have major impact on our life.The main goal of this project is to develop efficient numerical algorithms to capture rare events in infinite dimensional systems. We will integrate the techniques for numerical solution of partial differential equations, such as finite element method, reduced basis method, etc, (for the space-time dimension), with the ideas from large deviation theory, statistics, and deep learning (for the random dimension). When the large deviation principle is applicable, we will consider numerical solution of a nonlocal variational problem to seek the most probable event. The algorithm will be developed and analyzed in the framework of finite element method and calculus of variation. When the large deviation principle is not applicable, we will develop a strategy to seamlessly couple the reduced-order modeling and the generative models from deep learning, based on which a more general cross entropy method will be constructed for rare event simulations.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.

基于偏微分方程的数学建模方法在工程和科学应用中得到了广泛的应用，它已经成为人类理解人类活动和技术发展所产生的各种现象的重要工具之一。随机偏微分方程组是对偏微分方程组的推广，它考虑了现实中普遍存在的不确定性。在这个项目中，我们致力于模拟和量化随机偏微分方程中的罕见事件，它可以模拟一些重要的现象，如气候变化、异常海浪、异常天气等，这些现象可能很少发生，但对我们的生活有重大影响。本项目的主要目标是开发高效的数值算法来捕获无限维系统中的罕见事件。我们将结合大偏差理论、统计学和深度学习的思想(对于随机维度)，整合偏微分方程组的数值求解技术，如有限元方法、约简基方法等(针对时空维度)。当大偏差原理适用时，我们将考虑非局部变分问题的数值解，以寻求最可能的事件。该算法将在有限元方法和变分的框架内进行开发和分析。当大偏差原则不适用时，我们将开发一种策略，将深度学习的降阶建模和生成模型无缝结合，在此基础上构建更通用的交叉熵方法，用于罕见事件模拟。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Deep density estimation via invertible block-triangular mapping

• DOI: 10.1016/j.taml.2020.01.023
• 发表时间: 2020-03
• 期刊: Theoretical and Applied Mechanics Letters
• 影响因子: 3.4
• 作者: Keju Tang;X. Wan;Qifeng Liao

A Minimum Action Method for Dynamical Systems with Constant Time Delays

• DOI: 10.1137/20m1349163
• 发表时间: 2021
• 期刊: SIAM J. Sci. Comput.
• 作者: X. Wan;Jiayu Zhai

Adaptive Deep Density Approximation for Fractional Fokker–Planck Equations

• DOI: 10.1007/s10915-023-02379-z
• 发表时间: 2022-10
• 期刊: Journal of Scientific Computing
• 影响因子: 2.5
• 作者: Li Zeng;X. Wan;Tao Zhou

VAE-KRnet and its applications to variational Bayes

• DOI: 10.4208/cicp.oa-2021-0087
• 发表时间: 2020-06
• 期刊: ArXiv
• 作者: X. Wan;Shuangqing Wei

Adaptive deep density approximation for Fokker-Planck equations

• DOI: 10.1016/j.jcp.2022.111080
• 发表时间: 2021-03
• 期刊: J. Comput. Phys.
• 作者: Keju Tang;X. Wan;Qifeng Liao