Collaborative Research: Theory, computation and applications of parameterized Wasserstein gradient and Hamiltonian flows

近年来，非线性降阶模型对科学计算，特别是对深度神经网络(DNN)的研究产生了重大影响。曾经被认为难以解决的科学计算问题，例如求解高维偏微分方程组(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获得稳健和可靠的结果需要对其数学理论和计算有比当前文献所提供的更深入的理解。这个项目的目的是开发新的公式和严格的误差估计来使用DNN来求解一类重要的偏微分方程组，称为Wasserstein几何流，在DNN参数空间中。这项研究有望推进这些基于DNN的方法的数学基础，并使有效的计算算法能够在实践中求解这些偏微分方程组。该项目也自然提供了研究课题，以培养这一跨学科领域的下一代数学家和工程师。具有最优传输度量的概率分布空间中的几何流，即所谓的瓦瑟斯坦流形，在科学和工程中是普遍存在的。沃瑟斯坦梯度流和哈密顿流是这类流中最普遍的两种类型。这个项目开发了一个新的框架来分析和计算DNN参数空间中的瓦瑟斯坦梯度和哈密顿流。主要目标包括：(I)通过前推映射和回拉Wasserstein度量建立参数化的Wasserstein子流形及其数学基础；(Ii)发展参数化Wasserstein几何流动的计算高效的公式，并在Wasserstein空间中进行收敛分析和误差估计；(Iii)将新的公式与几个经典的偏微分方程组(包括Schrödinger方程和Fokker-Planck方程)联系起来。参数化瓦瑟斯坦梯度流和哈密顿流是它们在瓦瑟斯坦流形上对应的有限维非传统近似。他们提出了新的公式，具有近似保证，是通用的，并服从于实际的计算算法。这些结果也可以扩展到应用数学中涉及密度演化的其他主题，如生成模型、平均场控制和平均场博弈。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。

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.