Hopfield Neural Network (HNN) is a neural network with cyclic and recursive characteristics, combined with storage and binary systems. Invented by John Hopfield in 1982. For a Hopfield neural network, the key is to determine its weight under stable conditions . Hopfield neural networks are divided into discrete and continuous types. The main difference lies in the activation function.

Hopfield神经网络(HNN)是具有循环和递归特征的神经网络，结合了存储和二进制系统。由John Hopfield于1982年发明。对于Hopfield神经网络，关键是要在稳定的条件下确定其重量。 Hopfield神经网络分为离散类型和连续类型。主要区别在于激活功能。

The Hopfield Neural Network (HNN) provides a model that simulates human memory. It has a wide range of applications in artificial intelligence, such as machine learning, associative memory, pattern recognition, optimized calculation, VLSI and parallel realization of optical devices.

Hopfield神经网络(HNN)提供了一个模拟人类记忆的模型。它在人工智能中具有广泛的应用，例如机器学习，关联存储器，模式识别，优化计算，VLSI和光学设备的并行实现。

网络结构 (Network Structure)

The HopField network is a cyclic neural network with feedback connections from output to input. After the signal is input, the state of the neuron will continue to change over time, and finally converge or periodically oscillate.

HopField网络是一个循环神经网络，具有从输出到输入的反馈连接。输入信号后，神经元的状态将随着时间的推移而不断变化，并最终收敛或周期性地振荡。

Suppose there are a total of N neurons, x_i is the input of the i-th neuron, w_{ij} is the weight between neuron i and j , θ_i is the threshold of the i-th neuron, if the i-th neuron is in the state y(i, t) at time t then we have the following recurrence formula:

假设总共有N个神经元，x_i是第i个神经元的输入，w_ {ij}是神经元i和j之间的权重，θ_i是第i个神经元的阈值(如果第i个神经元)在时间t处于y(i，t)状态，那么我们有以下递推公式：

For simplicity, we consider a discrete HopField network, that is, f(x) is the sign function.

为简单起见，我们考虑一个离散的HopField网络，即f(x)是符号函数。

We introduce the concept of energy and define the increment of energy as:

我们介绍能量的概念，并将能量的增量定义为：

And,

和，

As long as the state y of neuron i changes whether from 1 to -1 or from -1 to 1, the change value in energy Δ E will be negative. When the HopField network reaches stability, the energy function is minimal.

只要神经元i的状态y从1变为-1或从-1变为1，能量ΔE的变化值将为负。当HopField网络达到稳定状态时，能量函数最小。

We define the energy of neuron i at time t as:

我们将在时间t的神经元i的能量定义为：

The total energy is:

总能量为：

Hopfield网络状态更改的示例 (Example of Hopfield network status change)

John Hopfield found that if the network connection is symmetric (wij=wji) and neurons have no self-feedback (wii=0), the network has a global energy function, and the correct network update rules make this energy function converge to a minimum. The update rules are:

约翰·霍普菲尔德(John Hopfield)发现，如果网络连接是对称的(wij = wji)，并且神经元没有自反馈(wii = 0)，则网络具有全局能量函数，并且正确的网络更新规则会使该能量函数收敛为最低。更新规则是：

Let’s use a specific example to illustrate that the neurons of the Hopfield network start from a random state, update one neuron at a time in a serialized manner, and finally reach a state of energy minimum.

让我们使用一个特定的示例来说明Hopfield网络的神经元从随机状态开始，以序列化的方式一次更新一个神经元，最后达到能量最小的状态。

In the initial state, only three neurons are activated and the total energy is -7.

在初始状态下，只有三个神经元被激活，总能量为-7。

After updating 5 neurons, the total energy of the network reaches a minimum, and the network is in a stable state.

更新5个神经元后，网络的总能量达到最小值，并且网络处于稳定状态。

In addition to updating the status of each node in the sequence of serialization, the convergence can also be updated in a synchronous manner .

除了按照序列化顺序更新每个节点的状态之外，收敛还可以以同步方式更新。

The final stable state of the network is called an attractor.

网络的最终稳定状态称为吸引子。

学习规则 (Learning rules)

There are many learning methods, the most common is Hebb learning rule.

学习方法很多，最常见的是赫布学习法则。

To learn a pattern p=[p_1,…,p_n] where pij=-/+1 .

要学习模式p = [p_1，…，p_n]，其中pij =-/ + 1。

Source资源

Given n interconnected neurons, the learning process regulation principle is that if two neurons i and j are excited at the same time, then the connection between them should be strengthened :

给定n个相互连接的神经元，学习过程的调节原理是，如果同时激发两个神经元i和j，则应加强它们之间的联系：