Waifu2x is not actually the first or only image scaler to use neural networks - NNEDI3[1], an Avisynth[2] filter used for deinterlacing can also do really nice image upscaling (and it's a lot faster than waifu2x). Here's an example of what it can do to the images in the blogpost:
Image 1: http://i.imgur.com/4cXr51v.png
Image 2: http://i.imgur.com/PZAXeM8.png
It doesn't come with any noise reduction, but nothing stops you from doing that separately from the upscaling process itself, and that way you should be able to control it better anyway (I find the reduction options provided by waifu2x really aggressive even with the low setting, it just kills tons of detail).
As a sidenote, when talking about something like image scaling, it would be a good idea to avoid saying something like "image scaled 2x (normally)" as there are lots of ways to scale images and what's "normal" can vary a lot depending on what you're using.