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.
As artificial intelligence (AI)-generated content proliferates, models are increasingly trained on their own outputs, risking progressive degradation or collapse. In this article, we provide the first positive, rigorous theoretical results, to the best of our knowledge, showing that under model-agnostic mild conditions, the model converges to the true data-generating distribution. The convergence rate is the minimum of the model's intrinsic rate and the fraction of real data at each training iteration, revealing a phase transition between data-limited and model-limited regimes. We further show that, for biased real data, correcting the bias prevents the persistence and amplification of early bias over training iteration. Extensive experiments across simulations, real images and texts validate our theoretical framework, establishing quantitative conditions for long-term AI stability in contaminated environments.
Complex physical systems, from supersonic turbulence to the macroscopic structure of the universe, are governed by continuous multiscale dynamics. While modern machine learning architectures excel at mapping the high-dimensional observables of these systems, it remains unclear whether they internalize the governing physical laws or merely interpolate discrete statistical correlations. Standard Explainable AI (XAI) architectures, particularly perturbation-based and gradient-saliency methods, rely on pixel-wise perturbations, which generate unphysical artifacts and push inputs off the valid empirical distribution. To resolve this, we introduce a diagnostic framework driven by Constrained Diffusion Decomposition (CDD), a diffusion-based multiscale data decomposition algorithm that enables physically constrained data generation and model evaluation via scale-aware modifications. Applying this framework to a Denoising Diffusion Probabilistic Model (DDPM), we execute deterministic interventions directly within the continuous, CDD-based scale space. We demonstrate that under moderate physical perturbations, the unconstrained generative model exhibits localized structural freezing and non-linear instability rather than continuous PDE-like responses. The network fails to maintain cross-scale continuity, causing the generative trajectory to diverge when pushed into unseen physical states. By synthesizing a continuum of physically coherent states, this scale-informed methodology establishes a controlled test ground to evaluate algorithmic vulnerabilities, providing the rigorous physical constraints necessary for future architectures to respect the multiscale causality of the natural universe.
Chaos arises in many complex dynamical systems, from weather to power grids, but is difficult to accurately model with data-driven methods such as machine learning emulators. While emulators are promising tools for accelerating simulations and solving inverse problems, they still struggle to learn chaotic dynamics, where sensitivity to initial conditions renders exact long-term forecasts infeasible, especially given noisy data. Recent work instead trains emulators to match the statistical properties of chaotic attractors, but these approaches often rely on handcrafted summary statistics or large, diverse multi-environment datasets. In this work, we propose a family of adversarial optimal transport objectives that can jointly learn high-quality summary statistics and a physically consistent emulator from a single noisy trajectory. We theoretically analyze and experimentally validate a Sinkhorn divergence formulation (2-Wasserstein) and a WGAN-style dual formulation (1-Wasserstein) of our approach. Numerical experiments across a variety of chaotic systems, including ones with high-dimensional spatiotemporal chaos, show that emulators trained using our proposed objectives have significantly improved long-term statistical fidelity.