中级Intermediate大纲Outline

多层神经网络

Neural Networks with Layers

一个神经元做不到的事，几个神经元先分工、再一个综合就能做到。下一步：多层网络怎么学。

What one neuron can’t do, a few neurons can: split the job, then combine. Next: how a layered network learns.

这一级正在开发，下面是大纲。

This level is still being built. Here is the outline.

感知机证明了：横线可以在任何一行时，一个神经元怎么都学不会，什么门槛都没用。

The perceptron showed that one neuron can never learn “a line in any row”, whatever its threshold.

第一层 3 个神经元，每个只管一行：这一行每格 1 分，门槛 2.5，三格全黑才亮。竖线只碰到一格，不亮。Layer one has 3 neurons, one per row: 1 point for each square in its row, threshold 2.5, so it lights up only when all three are black. A standing line touches just one square and stays dark.
第二层 1 个神经元：3 个灯有一个亮，就说“是”。这样“任意位置的线”全对。Layer two has 1 neuron: if any of the 3 lights is on, say yes. Now “lines anywhere” is all right.
XOR（两个格子正好一个黑）也一样：一个管“左黑右白”，一个管“右黑左白”，第二层有一个亮就亮。XOR (exactly one of two squares black) works the same way: one neuron for “left black, right white”, one for “right black, left white”, and a second layer that lights if either does.

感知机的规则只会改最后一层；前面那层错在哪里，它不知道。The perceptron rule can only fix the last layer; it can’t tell what the earlier layer got wrong.
把“亮 / 不亮”换成平滑的“亮几分”，就能算出每个分数该往哪边改、改多少，一层一层往回传。这叫反向传播（1986）。Replace “on or off” with a smooth “how bright”, and you can work out which way and how much to change every point, passing it back layer by layer. This is backpropagation (1986).
门槛也变成要学的数（偏置），总分离门槛多远变成“有百分之几的把握”。The threshold becomes something to learn too (the bias), and how far the total is from it becomes “how many percent sure”.

3 × 3 上摆 3 个隐藏神经元，孩子先手动给分，解决“任意位置的线”。Three hidden neurons on a 3 × 3 grid. Kids set the points by hand to solve “lines anywhere”.
再让它自己学，看隐藏神经元各自学成了什么：是不是正好一行一个？Then let it learn by itself and see what each hidden neuron becomes. One per row?