Solving multiscale PDEs is difficult in high dimensional and/or convection dominant cases. The Lagrangian computation, interacting particle method, is shown to outperform solving PDEs directly (Eulerian). Examples include computing effective diffusivities, KPP front speed, and asymptotic transport properties in topological insulators. However the particle simulation takes long before convergence and does not have a continuous model. In this regard, we introduce the DeepParticle methods, which learn the pushforward map from arbitrary distribution to IPM-generated distribution by minimizing the Wasserstein distance. In particular, we formulate an iterative scheme to find the transport map and prove the convergence. On the application side, in addition to KPP invariant measures, our method can also investigate the blow-up behavior in chemotaxis models.
Zhongjian Wang is a William H. Kruskal Instructor in the Department of Statistics at the University of Chicago.
“Applied analysis and numerical computation for physics and engineering problems have always been a passion of mine. There have been models developed for describing different systems basing on different problems and my research interests are to numerically simulate mathematical models and analyze the error in calculating the phenomena. During my Ph.D. study, I worked on calculating effective diffusivity in chaotic and random flows. Recently I am focusing on reduced-order models in the propagation of chaos and their probabilistic analysis. Related methods include, among others, POD, tensor-train, time-dependent PCE, and some machine learning algorithms.”