Nonlinear space-time model reduction in the frequency domain

TL;DR

This work extends spectral solution operator projection (SSOP) to nonlinear systems by encoding entire state trajectories with frequency-domain SPOD modes and projecting the implicit nonlinear solution onto this space-time basis. Nonlinearity is handled as a trajectory-dependent forcing, closed via DEIM hyper-reduction or sparse triadic interactions for quadratic terms, resulting in a nonlinear algebraic system for SPOD coefficients that is solved iteratively. Across nonlinear Ginzburg-Landau tests, the SSOP approach delivers two orders of magnitude lower error than POD-Galerkin at similar online cost, and nearly matches the SPOD-projection error with relatively few modes, demonstrating robustness to out-of-sample forcings and affine parameter changes. The method offers a principled, scalable way to exploit spatiotemporal coherence for accurate, efficient space-time ROMs in nonlinear settings, with clear pathways for affine-parameter extensions and broader applicability to PDEs with structured nonlinearities.

Abstract

We propose a space-time reduced-order model (ROM) for nonlinear dynamical systems, building upon previous work on linear systems. Whereas most ROMs are space-only in that they reduce only the spatial dimension of the state, the proposed method leverages an efficient encoding of the entire trajectory of the state on the time interval $[0,T]$, enabling significant additional reduction. Trajectories are encoded using SPOD modes, a spatial basis at each temporal frequency tailored to the structures that appear at that frequency. These modes have a number of properties that make them an ideal choice for space-time model reduction, including separability and near-optimality for long trajectories. We derive a system of algebraic equations involving the SPOD coefficients, forcing, and initial condition by projecting an implicit solution of the governing equations onto the set of SPOD modes in a space-time inner product. We therefore refer to the method as spectral solution operator projection (SSOP). The online phase of SSOP comprises solving this system for the SPOD coefficients, given the initial condition and forcing. We find that SSOP gives two orders of magnitude lower error than POD-Galerkin projection at the same number of modes and CPU time across a suite of tests, including ones that use out-of-sample forcings and affine parameter variation. In fact, the method is substantially more accurate even than the projection of the solution onto the POD modes, which is a lower bound for the error of any method based on a linear space-only encoding of the state.

Figures (12)

• Figure 1: The relative error for the SPOD and POD trajectory encoders as a function of $r$, the number of modes used.
• Figure 2: Spectral Solution Operator Projection: A system of nonlinear algebraic equations is obtained by performing a space-time projection of a linear solution operator onto the set of SPOD modes used to encode trajectories. The linear solution operator returns the solution to the linear system given the initial condition and forcing. Nonlinearities are handled in this framework as a state-dependent forcing. The system is solved via a fixed-point iteration, which may be viewed as a perturbation series around the linear solution.
• Figure 3: A trajectory of the (standard) Ginzburg-Landau system with $\mu_0 = 0.229$. The green lines demarcate the location of the forcing. The spatiotemporal structure enables space-time modes, like SPOD modes, to represent the trajectory far more efficiently than space-only modes.
• Figure 4: SPOD energies. (a) The energy not captured by the first $rN_\omega$ modes, calculated as the ratio of the sum of the energies of the excluded modes to the sum of the energies of all the modes. The dashed line corresponds to the $r$ value used to determine the cutoff in (b). (b) The energy of the included (red) and excluded (blue) modes as a function of $\omega$ for $r = 5$. Inclusion of a given mode ${\boldsymbol \psi} _{km}$ is determined by whether the associated energy is among the $rN_\omega$ largest energies over all frequencies i.e., whether $\lambda_{km} \geq \tilde{\lambda}_{(rN_\omega)}$ (see \ref{['eq:SPOD:rk']}). The cutoff energy $\tilde{\lambda}_{(rN_\omega)}$ is shown in orange. The number of modes at each frequency $r(\omega)$ is shown in green on the right axis; the mean is $r = 5$.
• Figure 5: Error as a function of time for the proposed method and for POD-G, both with $r=5$. The curves represent the mean error over the $30$ test trajectories, and the shaded regions indicate the standard deviation of these errors. The dashed curves show the respective projection errors, which serve as lower bounds for the two methods.