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 of unsteady fluid systems. 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 the use of intrinsic measurement coordinates to build nonlinear models, circumventing the well-known issue of continuous mode deformation associated with methods based on the proper orthogonal decomposition. This approach is demonstrated on several relevant systems in fluid dynamics with low-dimensional dynamics.
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