Abstract

This talk discusses how analytical and data-driven perspectives jointly contribute to understanding the low-dimensional geometry underlying complex dynamical systems. Building on concepts such as inertial and approximate inertial manifolds for dissipative equations, we show how modern manifold-learning and neural networks can complement classical analysis by discovering similar structures directly from data. Techniques including Diffusion Maps, autoencoders, and gray-box corrections yield interpretable reduced dynamics that remain faithful to the governing physics. Illustrative examples span dissipative partial differential equations, thin-film flows, and canonical nonlinear oscillators such as the Lorenz and Rössler systems. Extending this geometric view further, we demonstrate how Alternating Diffusion Maps identify shared and system-specific latent variables across heterogeneous sensors in multi-view settings. Together, these developments highlight a growing synthesis between mathematical analysis and data-driven modeling in revealing the hidden manifolds that organize complex dynamics.