You asked,

Has Topaz ever stated if there are different AI backing these programs? Do they have recommendations on using one or all three for one image?

Yes to both. Straight from the source, at Topaz Lab's support site (emphasis mine, and some text omitted):

When To Use DeNoise AI, Sharpen AI, and Gigapixel AI

One of the most common questions we receive here at Topaz Labs is when should you use each of our plugins? The truth is there isn't one "perfect way" to use our programs but a thorough understanding of each program's purpose will help you make an educated decision for your own unique workflow.

When To Use DeNoise AI

DeNoise AI is intended to be a dedicated noise reduction software. In particular, it allows you to reduce noise while retaining fine detail that would be lost in traditional noise reduction tools. While it has some sharpening capabilities, these are not the same robust AI sharpening capabilities in Sharpen AI to address concerns such as motion blur and missed focus. This is just a basic sharpening tool for when more complex sharpening is not needed, similar to what you would find in Lightroom or Photoshop.

As for when to use DeNoise AI, consider how noise is affected by your editing. Anytime you edit a photo, you alter the pixels in that photo. Your noise is found within those same pixels, and as you edit you are affecting both the image you photographed and the noise.

Most of the time, you can probably run DeNoise AI wherever you want in your workflow and it will work just fine, but in some cases, if you edit the noise too much, or if the noise is not recognized by the AI model, the noise reduction might fail. Remember, DeNoise AI's noise reduction models are trained on unedited noise.

When such a conflict is suspected or likely, you should run DeNoise AI early in the editing process before you've edited the noise itself.

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When To Use Sharpen AI

Sharpen AI is intended to address several concerns including motion blur/camera shake, missed focus, and out-of-focus images. It also has some minor noise reduction capabilities similar to what you would find in Lightroom or Photoshop, but it should be noted that the noise reduction feature in Sharpen AI is not the same AI-based, detail-preserving noise reduction found in DeNoise AI. It is simply in Sharpen AI as a minor workflow tool when more robust noise reduction is not needed.

As for when to use Sharpen AI, this depends on the photo itself. Traditionally, you would do most of your sharpening toward the end of an edit, but if you're correcting missed focus or motion blur, you'll probably want to use Sharpen AI earlier in your editing process so that your subsequent edits don't interfere with the process. For more generic sharpening to address "soft" images, you will probably want to do that toward the end like you might do in other workflows.

When to Use Gigapixel AI

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There are also some tools for noise reduction and blur removal, as well as color bleed reduction and face refinement built into Gigapixel as well. Like our other apps, these are not necessarily as robust as our dedicated apps for noise reduction and sharpening but can help speed up your workflow when such additional image improvement is not necessary. It should also be noted that Gigapixel has a model for animation and computer graphics images as well.

As for when to use Gigapixel, it is often used as a final workflow step but this can also depend on the original input image as well. If you want to enlarge an image you are happy with, use it last. If you need to enlarge the image in order to edit it properly, it's ok to use it first.