Stochastic Optimal Prediction for the Kuramoto-Sivashinsky Equation

TL;DR

The paper develops stochastic optimal prediction for the Kuramoto–Sivashinsky equation under underresolution by constructing a reduced model for the resolved Fourier modes using conditional expectations with respect to a data-driven initial measure. It derives a Mori–Zwanzig–type decomposition into Markovian, memory, and noise terms under a diagonal Gaussian density obtained via maximum likelihood, and analyzes two memory projections (linear and finite-rank) within the short-memory framework. Numerical experiments show that resolving all unstable modes with a linear projection yields accurate long-time predictions, while finite-rank projections can fail due to slowly decaying memory coefficients; leaving unstable modes unresolved leads to persistent memory effects and reduced predictive accuracy. The study highlights the trade-offs between projection choice, memory modeling, and the necessity of capturing long-time memory when underresolution intersects unstable dynamics, with implications for designing reduced models of complex PDEs.

Abstract

We examine the problem of predicting the evolution of solutions of the Kuramoto-Sivashinsky equation when initial data are missing. We use the optimal prediction method to construct equations for the reduced system. The resulting equations for the resolved components of the solution are random integrodifferential equations. The accuracy of the predictions depends on the type of projection used in the integral term of the optimal prediction equations and on the choice of resolved components. The novel features of our work include the first application of the optimal prediction formalism to a nonlinear, non-Hamiltonian equation and the use of a non-invariant measure constructed through inference from empirical data.

Figures (17)

• Figure 1: Evolution of the energy $E=\frac{1}{4\pi}\int_0^{2\pi} v_N^2(x,t)dx$.
• Figure 2: Log-log plot of the variances for different Fourier modes as computed by 10000 Monte-Carlo samples and as predicted by the Gaussian density.
• Figure 3: Autocorrelation for the unresolved mode with wavenumber 6 for different initial times. a) Real part, b) Imaginary part.
• Figure 4: Comparison of the autocorrelation for the unresolved mode with wavenumber 6 as computed from the full system and the moving average method. a) Real part, b) Imaginary part.
• Figure 5: Projection coefficient of the memory term for the equation for the resolved mode 1 on the resolved mode 4 for the first set of variables. a) Real part, b) Imaginary part.