The computational resources needed to simulate high-dimensional systems such as fluid flows limit our ability to track, forecast, and control their behavior in real time. To overcome these challenges, we construct simplified low-dimensional systems called Reduced-Order Models (ROMs) which approximate relevant behavior of the original system over a range of initial conditions, parameter values, and input signals. 

We aim to construct reliable ROMs using efficient computational methods and as little training data as possible. Often ROMs are obtained by projecting the original system onto a low-dimensional submanifold of the state space. A major focus of our research involves identifying appropriate manifolds, projections, and low-dimensional dynamics using data and governing equations. We develop efficient and scalable algorithms that leverage inherent problem structure to tackle these challenging computational tasks.

Operator learning means learning maps called "operators" between infinite-dimensional spaces of functions using data. Examples include solution operators for partial differential equations such as the operator mapping fluid flow fields at an initial time to the flow fields at later times obtained by solving the Navier-Stokes equations. Another important class of operators are the Koopman operators associated with any dynamical system. These are linear operators encoding the system's dynamics via the evolution of functions on the state space. Linearity of Koopman operators opens up the toolkits of spectral theory and operator semigroups for analysis of nonlinear systems, with the caveat that Koopman operators for most interesting systems are necessarily infinite-dimensional. 

Some of the questions guiding our work include: Can we use learned operators to speed up simulations and other engineering tasks? How much data is needed to learn a reliable approximation of an operator? Can physical information be used to impose useful constraints on operator learning architectures? How do we certify the accuracy of a learned operator?

In the physical sciences, we often have to learn a lot from a small amount of data. In order to enable data-efficient learning, the architectures of our machine learning models must incorporate constraints coming from our prior knowledge about the system of interest. Built-in symmetries and conservation properties allow models to be trained with less data and to extrapolate across known transformations of the input. Can symmetries be learned from data and/or used as a form of regularization? What is the sample complexity to learn a symmetric model? Furthermore, certain structured model architectures can be trained using significantly less data than others. What model structures are consistent with underlying problem physics and enable data-efficient learning?

Our goal is to determine what a system is doing from limited sensor measurements, and then to use this information to forecast and control future dynamics. Some of the questions are work seeks to address include: Where should we place sensors to gain as much information as possible about the system's state? Where should we place actuators to exert maximum influence with the least control effort? How can we efficiently assimilate sensor measurements with expensive model-based forecasts of high-dimensional systems? How can we reliably control such as system in real time? Our approaches to these problems draw on tools from model reduction, operator learning, and machine learning with structure.