Low-dimensional dynamical analysis of chaotic falling films: manifold learning and coherent structure

Due to the dissipative nature of the Navier–Stokes equations, the long-time dynamics of are expected to collapse onto a low-dimensional manifold, reducing the effective dimension compared to a full state space. On this manifold, the system's state can be accurately parameterized, enabling more compact representations and faster computations. In this study, we demonstrate that such a low-dimensional representation is achievable for a chaotic two-dimensional vertical falling film. Solutions to the film equation, obtained via pseudospectral methods, are processed using principal component analysis and mapped to a lower-dimensional space using autoencoders. The temporal evolution of these representations is then learned using neural ODE. We assess the predictive performance of the reduced-order model by estimating probability density functions, autocorrelation functions, and energy spectra, all of which show good agreement with the ground truth simulations. We leverage the reduced state space to identify exact coherent states that we converge in the DNS.