Data-Driven Discovery and Control of Complex Systems: Uncovering Interpretable and Generalizable Nonlinear Models

Accurate and efficient reduced-order models are essential to understand, predict, estimate, and control complex, multiscale, and nonlinear dynamical systems. These models should ideally be generalizable, interpretable, and based on limited training data. This work develops a general framework to discover the governing equations underlying a dynamical system simply from data measurements, leveraging advances in sparsity-promoting techniques and machine learning. The resulting models are parsimonious, balancing model complexity with descriptive ability while avoiding overfitting. This perspective, combining dynamical systems with machine learning and sparse sensing, is explored with the overarching goal of real-time closed-loop feedback control. First, we will discuss how it is possible to enforce known constraints, such as energy conserving quadratic nonlinearities in incompressible fluids, to “bake in” known physics. Next, we will demonstrate that higher-order nonlinearities can approximate the effect of truncated modes, resulting in more accurate models of lower order than Galerkin projection. Finally, we will discuss how to design sparse sensors and design models based on intrinsic measurements of the system.