从确定性混沌系统中单条轨迹学习生成模型，突破传统独立同分布假设
Generative Adversarial Learning from Deterministic Processes
- 用无限维生成对抗学习框架建模混沌系统演化轨迹
- 仅需一条确定性时间序列即可收敛至系统不变分布，收敛率由JS散度给出
- 为物理人工智能提供理论支持，适合研究非随机系统建模的学者
物理人工智能正成功应用于不遵循传统独立同分布（i.i.d.）样本范式的数据。事实上，物理人工智能常训练于非随机数据，这些数据源自湍流等混沌动力系统。我们以生成对抗网络（GAN）为例，解释其经验成功的理论基础——尽管在i.i.d.假设下其统计学习理论已被广泛理解。本文证明，在无限维生成对抗学习（GAL）模型下，仅需一个足够混沌的动力系统状态或其测量值的单条确定性时间序列，即可学习该系统的不变分布，并给出了基于杰恩-申松散度（Jensen-Shannon divergence）的收敛速率。
原文摘要 · Abstract (English)
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.
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