Machine Learning, Dynamical Systems and Control

PART IV: Reduced Order Models (ROMs)

The proper orthogonal decomposition (POD) is the SVD algorithm applied to partial differential equations (PDEs). As such, it is one of the most important dimensionality reduction techniques available to study complex, spatio-temporal systems. Such systems are typically exemplified by nonlinear partial differential equations that prescribe the evolution in time and space of the quantities of interest in a given physical, engineering and/or biological system. The success of the POD is related to the seemingly ubiquitous observation that in most complex systems, meaningful behaviors are encoded in low-dimensional patterns of dynamic activity. The POD technique seeks to take advantage of this fact in order to produce low-rank dynamical systems capable of accurately modeling the full spatio-temporal evolution of the governing complex system. Specifically, reduced order models (ROMs) leverage POD modes for projecting PDE dynamics to low-rank subspaces where simulations of the governing PDE model can be more readily evaluated. Importantly, the low-rank models produced by the ROM allow for significant improvements in computational speed, potentially enabling prohibitively expensive Monte-Carlo simulations of PDE systems, optimization over parametrized PDE systems, and/or real-time control of PDE-based systems.