Collaborative Research: Theory, computation and applications of parameterized Wasserstein gradient and Hamiltonian flows
合作研究:参数化 Wasserstein 梯度和哈密顿流的理论、计算和应用
基本信息
- 批准号:2307466
- 负责人:
- 金额:$ 14.23万
- 依托单位:
- 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:2023-07-01 至 2026-06-30
- 项目状态:未结题
- 来源:
- 关键词:
项目摘要
In recent years, nonlinear reduced-order models have demonstrated significant impact on scientific computing, particularly on deep neural networks (DNNs). Scientific computing problems that were once considered intractable, such as solving high dimensional partial differential equations (PDEs), became possible using DNNs which are scalable as the dimension increases. However, achieving robust and reliable results using DNNs requires a deeper understanding of their mathematical theory and computation than the current literature provides. This project aims to develop novel formulations and rigorous error estimates for using DNNs to solve an important class of PDEs, called the Wasserstein geometric flows, in the space of DNN parameters. This research is expected to progress the mathematical underpinnings of these DNN-based approaches and enable effective computational algorithms for solving these PDEs in practice. The project also naturally provides research topics to train the next generation of mathematicians and engineers in this interdisciplinary field. Geometric flows in the space of probability distributions equipped with the optimal transport metric, the so-called Wasserstein manifold, are ubiquitous in science and engineering. Wasserstein gradient and Hamiltonian flows are two of the most prevalent types of such flows. This project develops a novel framework to analyze and compute the Wasserstein gradient and Hamiltonian flows in the space of DNN parameters. The main objectives include (i) establishing parameterized Wasserstein sub-manifolds, via a push-forward map and pullback Wasserstein metric and their mathematical foundations; (ii) developing computationally efficient formulations for the parameterized Wasserstein geometric flows and conducting convergence analysis and error estimates in Wasserstein space; and (iii) connecting the new formulations to several classical PDEs including the Schrödinger and Fokker-Planck equations. The parameterized Wasserstein gradient and Hamiltonian flows are nontraditional, finite dimensional approximations of their counterparts on the Wasserstein manifold. They present new formulations that have approximation guarantees, are versatile, and are amenable to practical computational algorithms. These results can also be extended to other topics in applied mathematics involving density evolutions, such as generative models, mean-field control, and mean-field games.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.
近年来,非线性降阶模型对科学计算,特别是深度神经网络(DNN)产生了重大影响。曾经被认为是棘手的科学计算问题,例如求解高维偏微分方程(PDE),使用DNN变得可能,DNN随着维度的增加而可扩展。然而,使用DNN实现鲁棒和可靠的结果需要比现有文献更深入地理解其数学理论和计算。该项目旨在开发新的公式和严格的误差估计,用于使用DNN在DNN参数空间中解决一类重要的偏微分方程,称为Wasserstein几何流。这项研究有望推进这些基于DNN的方法的数学基础,并在实践中实现有效的计算算法来解决这些偏微分方程。该项目也自然提供了研究课题,以培养下一代数学家和工程师在这个跨学科领域。 在概率分布空间中的几何流配备了最佳的传输度量,所谓的Wasserstein流形,在科学和工程中无处不在。Wasserstein梯度流和Hamiltonian流是这类流中最普遍的两种。该项目开发了一种新的框架来分析和计算DNN参数空间中的Wasserstein梯度和Hamilton流。主要目标包括:(i)通过推进映射和拉回Wasserstein度量及其数学基础建立参数化Wasserstein子流形;(ii)在Wasserstein空间中建立参数化Wasserstein几何流的计算有效公式,并进行收敛性分析和误差估计;以及(iii)将新公式与包括薛定谔方程和福克-普朗克方程在内的几个经典偏微分方程联系起来。参数化的Wasserstein梯度流和Hamilton流是Wasserstein流形上的非传统的有限维近似。他们提出了新的配方,有近似保证,是通用的,并符合实际的计算算法。这些结果也可以扩展到其他主题的应用数学涉及密度的演变,如生成模型,平均场控制,平均场game.This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。
项目成果
期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
High-Dimensional Optimal Density Control with Wasserstein Metric Matching
- DOI:10.1109/cdc49753.2023.10384042
- 发表时间:2023-07
- 期刊:
- 影响因子:0
- 作者:Shaojun Ma;Mengxue Hou;X. Ye;Haomin Zhou
- 通讯作者:Shaojun Ma;Mengxue Hou;X. Ye;Haomin Zhou
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Xiaojing Ye其他文献
LAMA-Net: A Convergent Network Architecture for Dual-Domain Reconstruction
- DOI:
10.1007/s10851-025-01249-7 - 发表时间:
2025-05-13 - 期刊:
- 影响因子:1.500
- 作者:
Chi Ding;Qingchao Zhang;Ge Wang;Xiaojing Ye;Yunmei Chen - 通讯作者:
Yunmei Chen
“Returning beyond cancer”—a journey of professional reinvention for nurses
- DOI:
10.1007/s00520-025-09467-w - 发表时间:
2025-04-25 - 期刊:
- 影响因子:3.000
- 作者:
Qingyi Xue;Wenjing Xu;Xulu Wang;Xiaojing Ye;Wanting Hong;Qianqian Chen;Xin Lu;Xiaolei Wang;Chunmei Zhang - 通讯作者:
Chunmei Zhang
GENE THERAPY OF SYSTEMIC LUPUS ERYTHEMATOSUS IN NZB/W F1 MICE
NZB/W F1 小鼠系统性红斑狼疮的基因治疗
- DOI:
- 发表时间:
2005 - 期刊:
- 影响因子:0
- 作者:
Xiaojing Ye - 通讯作者:
Xiaojing Ye
Plexin-A1 expression in the inhibitory neurons of infralimbic cortex regulates the specificity of fear memory in male mice
边缘下皮层抑制性神经元中 Plexin-A1 的表达调节雄性小鼠恐惧记忆的特异性
- DOI:
10.1038/s41386-021-01177-1 - 发表时间:
2021-09 - 期刊:
- 影响因子:7.6
- 作者:
Xin Cheng;Yan Zhao;Shuyu Zheng;Panwu Zhao;Jin-lin Zou;Wei-Jye Lin;Wen Wu;Xiaojing Ye - 通讯作者:
Xiaojing Ye
Neural Control of Parametric Solutions for High-Dimensional Evolution PDEs
高维演化偏微分方程参数解的神经控制
- DOI:
10.1137/23m1549870 - 发表时间:
2024 - 期刊:
- 影响因子:0
- 作者:
Nathan Gaby;Xiaojing Ye;Haomin Zhou - 通讯作者:
Haomin Zhou
Xiaojing Ye的其他文献
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{{ truncateString('Xiaojing Ye', 18)}}的其他基金
Collaborative Research: Algorithms for Learning Regularizations of Inverse Problems with High Data Heterogeneity
合作研究:高数据异质性逆问题的学习正则化算法
- 批准号:
2152960 - 财政年份:2022
- 资助金额:
$ 14.23万 - 项目类别:
Continuing Grant
ATD: Algorithms for Point Processes on Networks for Threat Detection
ATD:用于威胁检测的网络点处理算法
- 批准号:
1925263 - 财政年份:2019
- 资助金额:
$ 14.23万 - 项目类别:
Standard Grant
Collaborative Research: Prediction, Optimization and Control for Information Propagation on Networks: A Differential Equation and Mass Transportation Based Approach
合作研究:网络信息传播的预测、优化和控制:基于微分方程和大众运输的方法
- 批准号:
1620342 - 财政年份:2016
- 资助金额:
$ 14.23万 - 项目类别:
Standard Grant
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