Generative learning for nonlinear dynamics

Modern generative machine learning models are able to create realistic outputs far beyond their training data, such as photorealistic artwork, accurate protein structures or conversational text. These successes suggest that generative models learn to effectively parametrize and sample arbitrarily complex distributions. Beginning half a century ago, foundational works in nonlinear dynamics used tools from information theory for a similar purpose, namely, to infer properties of chaotic attractors from real-world time series. This Perspective article aims to connect these classical works to emerging themes in large-scale generative statistical learning. It focuses specifically on two classical problems: reconstructing dynamical manifolds given partial measurements, which parallels modern latent variable methods, and inferring minimal dynamical motifs underlying complicated data sets, which mirrors interpretability probes for trained models. Generative machine learning models seek to approximate and then sample the probability distribution of the data sets on which they are trained. This Perspective article connects these methods to historical studies of information processing and attractor geometry in nonlinear systems.