Ignore the "experts" part, it misleads a lot of people [0]. There is no explicit specialization in the most popular setups, it is achieved implicitly through training. In short: MoEs add multiple MLP sublayers and a routing mechanism after each attention sublayer and let the training procedure learn the MLP parameters and the routing parameters.

In a longer, but still rough, form...

How these transformers work is roughly:

``` x_{l+1} = mlp_l(attention_l(x_l)) ```

where `x_l` is the hidden representation at layer l, `attention_l` is the attention sublayer at layer l, and `mlp_l` is the multilayer perceptron at sublayer l.

This MLP layer is very expensive because it is fully connected (i.e. every input has a weight to every output). So! MoEs instead of creating an even bigger, more expensive MLP to get more capability, they create K MLP sublayers (the "experts") and a router that decides which MLP sublayers to use. This router spits out an importance score for each MLP "expert" and then you choose the top T MLPs and do an average weighed on importance, so roughly:

``` x_{l+1} = \sum_e mlp_{l,e}(attention_l(x_l)) * importance_score_{l, e} ```

where the `importance_score_{l, e}` is the score computed by the router at layer l for "expert" e. That is, `importance_score_{l} = attention_l(x_l)`. Note that here we are adding all experts, but in reality we choose the top T, often 2, and use that.

[0] some architectures do, in fact, combine domain experts to make a greater whole, but not the currently popular flavor