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A model for the dimension of the inertial manifold of the Kuramoto-Sivashinsky equation

  • Melisa Y. Vinograd
  • Patricio Clark di Leoni
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Abstract

We propose a model, based on a spectral energy balance argument, that provides an estimate of the dimension of the inertial manifold of the one-dimensional Kuramoto-Sivashinsky equation (KSE). By identifying the wavenumber at which energy injection and dissipation balance out, we obtain a cutoff scale kd=√(5/(3ν)) that we use to calculate the number of dynamically active Fourier modes. This leads to an explicit expression dM(L,ν) for the dimension of an effective manifold spanned by these modes, which we take as our estimate of the inertial manifold dimension and which scales extensively with domain size. Direct numerical simulations are consistent with the estimated spectral cutoff and its scaling with viscosity. Using convolutional autoencoders as well as Lyapunov-based estimates, we show that the minimal latent dimension required for accurate reconstruction coincides with dM across a range of (L,ν).

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Cite

BibTeXRIS
BibTeX
@misc{vinograd2026model,
  title = {{A model for the dimension of the inertial manifold of the Kuramoto-Sivashinsky equation}},
  author = {Melisa Y. Vinograd and Patricio Clark di Leoni},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41683},
  eprint = {2610.41683},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41683},
}
RIS
TY  - UNPB
TI  - A model for the dimension of the inertial manifold of the Kuramoto-Sivashinsky equation
AU  - Melisa Y. Vinograd
AU  - Patricio Clark di Leoni
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41683
UR  - https://arxiv.org/abs/2610.41683
AB  - We propose a model, based on a spectral energy balance argument, that provides an estimate of the dimension of the inertial manifold of the one-dimensional Kuramoto-Sivashinsky equation (KSE). By identifying the wavenumber at which energy injection and dissipation balance out, we obtain a cutoff scale k_d=√(5/(3ν)) that we use to calculate the number of dynamically active Fourier modes. This leads to an explicit expression d_M(L,ν) for the dimension of an effective manifold spanned by these modes, which we take as our estimate of the inertial manifold dimension and which scales extensively with domain size. Direct numerical simulations are consistent with the estimated spectral cutoff and its scaling with viscosity. Using convolutional autoencoders as well as Lyapunov-based estimates, we show that the minimal latent dimension required for accurate reconstruction coincides with d_M across a range of (L,ν).
ER  -