基于信息几何和归纳推理的动力系统概率复杂性刻画
Complexity Characterization in a Probabilistic Approach to Dynamical Systems Through Information Geometry and Inductive Inference
摘要：
信息几何技术和归纳推理方法在解决经典和量子物理中感兴趣的计算问题方面有很大的希望，特别是在弯曲统计流形上的概率描述方面，动力系统的复杂性表征。在这篇文章中，我们研究的可能性，描述的宏观行为的复杂系统的微观自由度的统计结构，通过使用统计归纳推理和信息几何。我们回顾了统计流形上的最大相对熵（MrE）形式主义和混沌的信息几何动力学方法（IGAC）的理论结构。特别着重描述了截面曲率、雅可比场强度和信息几何动力学熵（IGE）所起的作用。这些数量作为强大的信息几何复杂性措施的信息约束动力学与任意混沌和定期系统定义的统计流形上。最后，这些信息几何技术的应用，几个理论模型。
英文摘要：
Information geometric techniques and inductive inference methods hold great promise for solving computational problems of interest in classical and quantum physics, especially with regard to complexity characterization of dynamical systems in terms of their probabilistic description on curved statistical manifolds. In this article, we investigate the possibility of describing the macroscopic behavior of complex systems in terms of the underlying statistical structure of their microscopic degrees of freedom by use of statistical inductive inference and information geometry. We review the Maximum Relative Entropy (MrE) formalism and the theoretical structure of the information geometrodynamical approach to chaos (IGAC) on statistical manifolds. Special focus is devoted to the description of the roles played by the sectional curvature, the Jacobi field intensity and the information geometrodynamical entropy (IGE). These quantities serve as powerful information geometric complexity measures of information-constrained dynamics associated with arbitrary chaotic and regular systems defined on the statistical manifold. Finally, the application of such information geometric techniques to several theoretical models are presented.
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