Chris Olah has a great blog post on how topology relates to machine learning ("machine learning untangles highly kneaded spaces").

I will let him summarize:

While it is challenging to understand the behavior of deep neural networks in general, it turns out to be much easier to explore low-dimensional deep neural networks – networks that only have a few neurons in each layer. In fact, we can create visualizations to completely understand the behavior and training of such networks. This perspective will allow us to gain deeper intuition about the behavior of neural networks and observe a connection linking neural networks to an area of mathematics called topology.

 

A number of interesting things follow from this, including fundamental lower-bounds on the complexity of a neural network capable of classifying certain datasets.

His blog also has posts on other specific types of deep neural networks such as "convolutional neural networks", but I haven't read those.

Chris Olah has a great blog post on how topology relates to machine learning ("machine learning untangles highly kneaded spaces").

I will let him summarize:

While it is challenging to understand the behavior of deep neural networks in general, it turns out to be much easier to explore low-dimensional deep neural networks – networks that only have a few neurons in each layer. In fact, we can create visualizations to completely understand the behavior and training of such networks. This perspective will allow us to gain deeper intuition about the behavior of neural networks and observe a connection linking neural networks to an area of mathematics called topology.

A number of interesting things follow from this, including fundamental lower-bounds on the complexity of a neural network capable of classifying certain datasets.

His blog also has posts on other specific types of deep neural networks such as "convolutional neural networks", but I haven't read those.