神经网络的统计力学(英文版)

本书涵盖了用于理解神 经网络原理的必要统计力学 知识，包括复本方法、空腔 方法、平均场近似、变分法 、随机能量模型、Nishimori 条件、动力学平均场理论、 对称性破缺、随机矩阵理论 等，同时详细描述了监督学 习、无监督学习、联想记忆 网络、感知器网络、随机循 环网络等神经网络及其功能 的物理模型以及解析理论， 通过简洁的模型展示了神经 网络原理的数学美和物理深 度，介绍了相关历史并展望 了未来研究的重要课题，可 供对神经网络原理感兴趣的 学生、研究人员以及工程师 参考使用。

1 Introduction References 2 Spin Glass Models and Cavity Method 2.1 Multi-spin Interaction Models 2.2 Cavity Method 2.3 From Cavity Method to Message Passing Algorithms References 3 Variational Mean-Field Theory and Belief Propagation 3.1 Variational Method 3.2 Variational Free Energy 3.2.1 Mean-Field Approximation 3.2.2 Bethe Approximation 3.2.3 From the Bethe to Naive Mean-Field Approximation 3.3 Mean-Field Inverse Ising Problem References 4 Monte Carlo Simulation Methods 4.1 Monte Carlo Method 4.2 Importance Sampling 4.3 Markov Chain Sampling 4.4 Monte Carlo Simulations in Statistical Physics 4.4.1 Metropolis Algorithm 4.4.2 Parallel Tempering Monte Carlo References 5 High-Temperature Expansion 5.1 Statistical Physics Seting 5.2 High-Temperature Expansion 5.3 Properties of the TAP Equation References 6 Nishimori Line 6.1 Model Setting 6.2 Exact Result for Internal Energy 6.3 Proof of No RSB Effects on the Nishimori Line References 7 Random Energy Model 7.1 Model Setting 7.2 Phase Diagram References 8 Statistical Mechanical Theory of Hopfield Model 8.1 Hopfield Model 8.2 Replica Method 8.2.1 Replica-Symmetric Ansatz 8.2.2 Zero-Temperature Limit 8.3 Phase Diagram 8.4 Hopfield Model with Arbitrary Hebbian Length 8.4.1 Computation of the Disorder-Averaged Free Energy 8.4.2 Derivation of Saddle-Point Equations 8.4.3 Computation Transformation to Solve the SDE 8.4.4 Zero-Te mperature Limit References 9 Replica Symmetry and Replica Symmetry Breaking