Generative Adversarial Learning from Deterministic Processes

2026-05-18

AI Summary

This work addresses the limitation of conventional generative adversarial networks (GANs), which rely on the independent and identically distributed (i.i.d.) assumption and thus struggle with deterministic, non-random time series generated by chaotic dynamical systems. The authors propose a novel framework based on infinite-dimensional generative adversarial learning that, for the first time, provides theoretical guarantees for GANs in a non-i.i.d. setting. Specifically, they demonstrate that the invariant distribution of a chaotic system can be recovered from a single deterministic trajectory, establishing an explicit convergence rate in terms of Jensen–Shannon divergence. By integrating concepts from chaotic dynamical systems, statistical learning theory, and infinite-dimensional GAN analysis, this study rigorously establishes the feasibility of learning invariant measures from a single sample path and quantifies the convergence speed of generative models toward the true underlying distribution.

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.

Problem

• Generative Adversarial Networks
• Deterministic Processes
• Chaotic Dynamical Systems
• Invariant Distribution
• Non-i.i.d. Data

Innovation

• Generative Adversarial Learning
• Chaotic Dynamical Systems
• Invariant Distribution
• Non-i.i.d. Data
• Jensen-Shannon Divergence