高维混沌系统中学习的统计物理学
Statistical physics of learning in high-dimensional chaotic systems
AI总结：
本文以高维混沌系统为范式模型，研究Hebbian驱动对混沌程度的调控以及FORCE算法训练系统重现周期函数的能力，并通过动力学平均场理论在热力学极限下给出精确分析。
摘要：
在许多复杂系统中，基本单元处于混沌环境中，需要调整自身策略以完成任务，即从环境中提取信息并控制其上的反馈回路。这类系统的一个主要例子是循环神经网络。在这种情况下，神经元之间的循环连接驱动混沌行为，而当学习发生时，系统对扰动的响应还应考虑其对网络自身动力学的反馈。在这项工作中，我们考虑一个高维混沌系统的抽象模型作为范式模型，并研究其动力学。我们在两种特定设置下研究该模型：Hebbian驱动和FORCE训练。在第一种情况下，我们表明Hebbian驱动可用于调节动力学中的混沌程度，这复现了最近在研究更具生物学真实性的循环神经网络模型时所获得的一些结果。在后一种情况下，我们表明该动力系统可以被训练以重现简单的周期函数。为此，我们考虑FORCE算法——最初为训练循环神经网络而开发——并将其适配到我们的高维混沌系统中。我们表明，训练时间越长，该算法将动力学驱动到越接近渐近吸引子。由于通过动力学平均场理论对动力学进行了精确分析，我们所有的结果在热力学极限下都是有效的。
英文摘要：
In many complex systems, elementary units live in a chaotic environment and need to adapt their strategies to perform a task, by extracting information from the environment and controlling the feedback loop on it. One of the main example of systems of this kind is provided by recurrent neural networks. In this case, recurrent connections between neurons drive chaotic behavior and when learning takes place, the response of the system to a perturbation should take into account also its feedback on the dynamics of the network itself. In this work, we consider an abstract model of a high-dimensional chaotic system as a paradigmatic model and study its dynamics. We study the model under two particular settings: Hebbian driving and FORCE training. In the first case, we show that Hebbian driving can be used to tune the level of chaos in the dynamics and this reproduces some results recently obtained in the study of more biologically realistic models of recurrent neural networks. In the latter case, we show that the dynamical system can be trained to reproduce simple periodic functions. To do this, we consider the FORCE algorithm -- originally developed to train recurrent neural networks -- and adapt it to our high-dimensional chaotic system. We show that this algorithm drives the dynamics close to an asymptotic attractor the larger the training time. All our results are valid in the thermodynamic limit thanks to an exact analysis of the dynamics through dynamical mean field theory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
