Generative learning for nonlinear dynamics
William Gilpin
Modern generative machine learning models demonstrate surprising ability 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 to infer
properties of chaotic attractors from time series, motivating the development
of algorithms for parametrizing chaos in real datasets. In this perspective, we
aim to connect these classical works to emerging themes in large-scale
generative statistical learning. We first consider classical attractor
reconstruction, which mirrors constraints on latent representations learned by
state space models of time series. We next revisit early efforts to use
symbolic approximations to compare minimal discrete generators underlying
complex processes, a problem relevant to modern efforts to distill and
interpret black-box statistical models. Emerging interdisciplinary works bridge
nonlinear dynamics and learning theory, such as operator-theoretic methods for
complex fluid flows, or detection of broken detailed balance in biological
datasets. We anticipate that future machine learning techniques may revisit
other classical concepts from nonlinear dynamics, such as transinformation
decay and complexity-entropy tradeoffs.