Manifold dimension identification for the Kuramoto-Sivashinsky equation via a spectral convolutional autoencoder.

Abstract

We present a spectral convolutional autoencoder framework to infer an integer, observation-conditioned estimate for the inertial-manifold dimension in the Kuramoto-Sivashinsky equation. Training on trajectory segments of temporal horizon τ, we analyze reconstruction-error scaling with latent dimension d through a local log-slope diagnostic γ(d,τ) and identify a robust horizon-induced change-point: once τ exceeds a mixing scale, the scaling gain localizes at a unique integer d^{*}(L) and remains stable for larger horizons. Across L={22,44,58,66,88,96}, the extracted d^{*} is extensive and is well described by a linear law, d^{*}≈0.366L. Moreover, d^{*} is largely insensitive to temporal-frequency truncation provided the energy-dominant low-frequency band is retained, indicating bandwidth sufficiency for stable dimension identification under the present observation scheme.