The neural net is just a math function, continuous even, fully differentiable in all input points. In order to "learn" anything we compute gradients towards the function parameters. They get "nudged" slightly towards a better response, and we do this billions of times. It's like carving a raw stone block into a complex scene. If you put your data into the system it flows towards the desired output because the right path has been engraved during training.

<rant> This explains a bit how neural nets work but from this to chatGPT is another whole leap. You'd have to assign some of the merits of the AI to the training data itself, it's not just the algorithm for learning, but what is being learned that matters. The neural net is the same, but using 1T tokens of text is making it smart. What's so magic about this data that it can turn a random init into a language programmable system? And the same language information make a baby into a modern human, instead of just another animal. </>

No, it's the loss function we differentiate. The input to the loss function are the network weights. The input to the network are samples and those we do not differentiate.

While it's true that we don't differentiate the input samples, we do differentiate the loss function's output with respect to each of the network weight. We use the chain rule to calculate each of these "gradients" and that process is known as backpropagation.

Same thing. You need everything to be differentiable in order to run gradient descent. To first approximation, training a neural network is just gradient descent.

Not an expert (so this could well be slightly off), but here is a physics analogy:

You're rolling a ball down a plank of wood and you want it to roll off the plank and continue to a certain spot on the ground. You know that if the ball rolls past the target, you have to angle the ramp lower. If it fails to reach the target, you angle the ramp higher. If it goes past by a lot, you make a big change to the ramp angle. If it goes past by a little, you make a small change to the ramp angle. In this way your error tells you something about the adjustment you need to make to the system to produce the desired output.

Think of a function from the ramp angle (network weights) to how far the ball lands from the target (the error). The derivative of this function tells you what kind of change in angle results in what kind of change in the error, and you can use that to update the ramp until you hit the target. To say that the model is differentiable is to say that it's possible to infer from the error what kind of changes you need to make to the weights.