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。

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.