Data-Driven Latent Dynamics Discovery with Deep Delay Autoencoders

Abstract

The analysis of natural physical systems is often constrained by incomplete measurements where only a subset of the underlying state variables can be observed. This challenge is especially evident in marine ecosystems, where complex, multi-scale interactions limit direct observation of all the interacting components. From a dynamical systems perspective, such partial observations can be interpreted as projections of an underlying dynamical object in state space, allowing for the reconstruction of system dynamics through delay-coordinate embedding techniques. In this thesis, we investigate data-driven approaches for recovering latent dynamics and modeling dynamical systems from partial and noisy observations. We combine information-theoretic and geometrical embedding methods with machine learning techniques, specifically leveraging deep autoencoder architecture to learn nonlinear coordinate transformations, coupled with the Sparse Identification of Nonlinear Dynamics (SINDy) framework to identify interpretable governing equations. To address sensitivity to noise, we introduce an ensemble-based extension, Ensemble SINDy, which improves the robustness and stability of model discovery. The proposed methodology is evaluated on both noisy synthetic systems and ecological reanalysis datasets based on satellite observations and numerical models. Results demonstrate that our approach can recover meaningful dynamical information from noisy and partially observed data while recognizing the challenges and limitations.