Frequency-domain nonlinear model reduction using SPOD modes

Spectral proper orthogonal decomposition (SPOD) has been shown to effectively identify large-scale spatiotemporal coherent structures in fluids systems that play a key role in the dynamics. We present a frequency-domain-based model reduction method that uses SPOD modes to represent the trajectory of the state and solves a system of algebraic equations for the SPOD coefficients given the initial condition and forcing. A significant advantage of this approach is that by leveraging spatiotemporal correlations, trajectories can be represented to orders-of-magnitude more accuracy using some number of SPOD mode coefficients than they can with the same number of POD mode coefficients. In previous work, we developed a model reduction technique to solve quickly and accurately for the SPOD coefficients in linear systems and found that the method achieved substantially lower error than POD-Galerkin and balanced truncation at the same CPU time. In this talk, we extend the method to nonlinear systems. The method selects the most energetic triadic interactions and uses them to compute the effect of the nonlinearity. In many systems, a small number of the triadic interactions account for most of the nonlinearity, so by excluding all but these high-energy interactions, the online time of the method is substantially reduced. In the case of a non-quadratic nonlinearity, the method uses a hyper-reduction technique to handle the nonlinear term. We show that in both cases, we are able to solve the algebraic system that results quickly and that we recover the accuracy afforded by the spatiotemporal trajectory representation.