NSF Award
2307466
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. <br/><br/>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.<br/><br/>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.
