Low-dimensional models from upper bound theory

A novel model reduction strategy for forced dissipative infinite-dimensional nonlinear dynamical systems is described. Unlike popular but empirical methods, this new approach does not require extensive data sets from experiments or direct numerical simulations of the governing field equations. Instead, truly predictive reduced dynamical models are constructed using Galerkin projection onto a priori eigenfunctions drawn from energy stability and upper bound theory. Within the context of porous medium convection, we show that these eigenfunctions contain information about boundary layers and other complex dynamic features and, thus, are well suited for the low-order description of highly nonlinear phenomena. Crucially, our analysis reveals the existence of a gap in the eigenvalue spectrum—even for strongly supercritical forcing conditions—enabling the identification of a rational truncation scheme. We demonstrate the efficacy of our approach via comparison with Fourier–Galerkin approximations of various orders.