Continuous dynamical systems are cornerstones of many scientific and engineering disciplines. While machine learning offers powerful tools to model these systems from trajectory data, challenges arise when these trajectories are captured as images, resulting in pixel-level observations that are discrete in nature. Consequently, a naive application of a convolutional autoencoder can result in latent coordinates that are discontinuous in time. To resolve this, we propose continuity-preserving convolutional autoencoders (CpAEs) to learn continuous latent states and their corresponding continuous latent dynamical models from discrete image frames. We present a mathematical formulation for learning dynamics from image frames, which illustrates issues with previous approaches and motivates our methodology based on promoting the continuity of convolution filters, thereby preserving the continuity of the latent states. This approach enables CpAEs to produce latent states that evolve continuously with the underlying dynamics, leading to more accurate latent dynamical models. Extensive experiments across various scenarios demonstrate the effectiveness of CpAEs.

The perfect optical vortex (POV) beam with the characteristic of the topological charge (TC)- independence radial profile is first considered to improve the spatial-mode-based underwater wireless optical communication (UWOC) performance. The evolution model of POV beams is proposed on the basis of Rytov approximation and the mutual coherence function to explore the impact of wave parameters on the orbital angular momentum (OAM) detection probability spectrum. The instantaneous crosstalk coefficients describing the POV-based UWOC systems are simulated by the multiple phase screen method with the signal coefficient modeled by a mixture Gamma-Gamma and normal distribution. A discrete and memoryless channel and a set of noncollaborative channels are modeled to obtain the capacity performances of the spatial-mode modulating and multiplexing systems, respectively. Results show the transmission quality of POV beams in oceanic turbulence is most related to the beam radius, while it is nearly free from the wavelength, TC, and radius-thickness ratio. Enjoying the advantage of the TC-independence beam radius, POV beams-based UWOC systems could achieve better bit error rate and capacity performances than the widely used Laguerre-Gauss beams when the higher-order TCs are adopted to modulate and multiplex information, which also confirms the superiority of POV beams in spatial-mode-based UWOC systems.