Exact and Optimal Quadratization of Nonlinear Finite-DimensionalNonautonomous Dynamical Systems

Quadratization of polynomial and nonpolynomial systems of ordinary differential equations (ODEs)is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reac-tion modeling, and mathematical analysis. A quadratization reveals new variables and structuresof a model, which may be easier to analyze, simulate, and control, and provides a convenient pa-rametrization for learning. This paper presents novel theory, algorithms, and software capabilitiesfor quadratization of nonautonomous ODEs. We provide existence results, depending on the reg-ularity of the input function, for cases when a quadratic-bilinear system can be obtained throughquadratization. We further develop existence results and an algorithm that generalizes the processof quadratization for systems with arbitrary dimension that retain the nonlinear structure when thedimension grows. For such systems, we provide dimension-agnostic quadratization. An example issemidiscretized PDEs, where the nonlinear terms remain symbolically identical when the discretiza-tion size increases. As an important aspect for practical adoption of this research, we extendedthe capabilities of theQBeesoftware towards both nonautonomous systems of ODEs and ODEswith arbitrary dimension. We present several examples of ODEs that were previously reported inthe literature, and where our new algorithms find quadratized ODE systems with lower dimensionthan the previously reported lifting transformations. We further highlight an important area ofquadratization: reduced-order model learning. This area can benefit significantly from working inthe optimal lifting variables, where quadratic models provide a direct parametrization of the modelthat also avoids additional hyperreduction for the nonlinear terms. A solar wind example highlightsthese advantages.