Generative Adversarial Learning from Deterministic Processes

TL;DR

This paper demonstrates that generative adversarial networks can learn the invariant distribution of chaotic dynamical systems from a single deterministic time series, extending GAN theory beyond i.i.d. data.

Contribution

It introduces an infinite-dimensional model of GANs capable of learning from deterministic chaos, providing explicit convergence rates for such systems.

Findings

GANs can learn invariant distributions of chaotic systems from single trajectories

Explicit convergence rates in Jensen-Shannon divergence are derived

Theoretical foundation extends GAN applicability to deterministic data

Abstract

Physical AI is being successfully applied to data which does not follow the traditional paradigm of independent and identically distributed (i.i.d.) samples. In fact, physical AI is often trained on data which is not random at all, and is instead derived from chaotic dynamical systems like turbulence. We aim to explain the empirical success of these methods using the example of generative adversarial networks (GANs), whose statistical learning theory under the i.i.d. assumption is generally well understood. We prove that it is possible, using an infinite-dimensional model of generative adversarial learning (GAL), to learn the invariant distribution of a sufficiently chaotic dynamical system from a single deterministically evolving time series of its states or measurements thereof, and give explicit rates for the convergence to the solution in terms of the Jensen-Shannon divergence.