Mohammad Farazmand

Spatiotemporal dynamics of many natural and engineering systems are described by continuum models. On the one hand, high-fidelity numerical simulation of these models is a valuable tool for analysis and prediction of the system. Conversely, experimental measurements are often limited to sparse spatial locations. This disparity impedes our ability to use the experimental observations as input to numerical simulations which demand high-resolution measurements of the system state. This issue is present in many scientific fields such as fluid dynamics, physical oceanography, and quantitative biology. The objective of the proposed program is to rectify this disconnect by developing a rigorous method that combines offline high-resolution simulations with online experimental observations in order to make parsimonious real-time predictions from sparse measurements of the system. The proposed framework consists of two stages: (i) First, we leverage offline simulations to learn the quantities that need to be measured experimentally, together with their optimal measurement locations. These are determined to maximize the accuracy of future predictions. (ii) Optimal real-time measurements and offline simulations are combined into a machine learning algorithm to infer the future state of the system. While the resulting framework will have broad applications, during this program we will focus on its applications to the prediction of extreme events, such as ocean rogue waves and tsunamis.

The dynamics of spatiotemporal systems are routinely described by time-dependent partial differential equations (PDEs). The solutions of these PDEs often exhibit time-varying localized structures, with sharp gradients, surrounded by regions of relative quiescence. Efficiently resolving these multiscale structures has been a long-standing challenge in scientific computing. Currently, there are two broad classes of methods for addressing this challenge: 1. Adaptive methods which dynamically evolve the spatial discretization so that the computational grid is refined around the localized structure and less so in the quiescent regions. 2. Multiresolution methods, such as wavelets, which encode various scales in the basis instead of the discretization. The proposed program will develop a new and computationally efficient method called shape-morphing modes. The main idea behind this method is to use a time-dependent basis of functions that automatically morph their shapes over time and space in order to efficiently resolve all scales. Being mesh-free, the proposed method substantially reduces the computational cost as compared to existing adaptive methods. Furthermore, since the modes adapt themselves to the solution of the PDE, far fewer modes are needed to resolve all scales. This significantly reduces the memory requirements, thus outperforming the existing multiresolution methods.