Parametric Reduced Order Models for the Generalized Kuramoto--Sivashinsky Equations

TL;DR

The work develops and evaluates parametric projection-based ROMs for the 1D KS and generalized KS equations across regimes ranging from spatio-temporal chaos to quasi-periodic behavior. It implements POD and POD-DEIM ROMs with diverse parameter-sampling strategies, highlighting the importance of including short-time transient and chaotic snapshots in training to extend predictive capability beyond the training set. The study shows that ROMs can reproduce short-time dynamics and long-time statistical properties (power spectra) but may struggle with long-term trajectory accuracy in laminar quasi-periodic states, suggesting that tensor-based or parameter-specific bases could improve universality. Overall, the results provide practical guidance for constructing ROMs that remain accurate across multiple dynamical regimes, with DEIM offering computational efficiency comparable to full projections.

Abstract

The paper studies parametric Reduced Order Models (ROMs) for the Kuramoto--Sivashinsky (KS) and generalized Kuramoto--Sivashinsky (gKS) equations. We consider several POD and POD-DEIM projection ROMs with various strategies for parameter sampling and snapshot collection. The aim is to identify an approach for constructing a ROM that is efficient across a range of parameters, encompassing several regimes exhibited by the KS and gKS solutions: weakly chaotic, transitional, and quasi-periodic dynamics. We describe such an approach and demonstrate that it is essential to develop ROMs that adequately represent the short-time transient behavior of the gKS model.

Figures (14)

• Figure 1: Simulations of the KS equation. Cumulative variance ratio, $C_i$, for different values of $L$ and the corresponding optimal reduced dimension. The optimal reduced dimension is selected when $C_i$ becomes less than $10^{-2}$.
• Figure 2: Averaged prediction times for trajectories of the gKS equation with different values of $\gamma$ using ROMs constructed using gKS data with a single value of $\gamma$. Upper left - ROM is constructed from simulations with $\gamma=0$ and $W=1$. Upper right, bottom left, bottom right - ROM is constructed from simulations with $\gamma=5$ with $W=25,100,250$, respectively. We consider 3 sets of different initial conditions with $J=3,8,22$, where $J$ is the number of non-zero Fourier wavenumbers at time $t=0$ (see eq. \ref{['eq:initial_condition']}).
• Figure 3: Averaged prediction times for trajectories of the gKS equation with different values of $\gamma$ using multi-valued ROMs constructed using gKS data with $K=5$ values of $\gamma$ and $W=1$. For each $\gamma$, we consider 3 sets of different initial conditions with $J=3,8,22$, where $J$ is the number of non-zero Fourier wavenumbers at time $t=0$ (see eq. \ref{['eq:initial_condition']}).
• Figure 4: Power Spectra (left) and a typical $L^2$ error for a single trajectory (right) for $\gamma = 0.1$.
• Figure 5: Comparison of an individual solution for FOM and $\text{ROM}_j$, $j=1,2,3$, for $\gamma = 0.1$.

Theorems & Definitions (1)

• Remark