I use a long term (e.g. "artificial neural network") more than once in the abstract of a paper. Is it best to introduce the abbreviation (e.g. "ANN") after the first occurrence, or repeat the non-abbreviated term each time?
It is almost certainly not "best" to introduce acronyms/abbreviate terms like "artificial neural network" in the abstract. I can think of two reasons to use acronyms/abbreviations in a manuscript. The first is to save words/space and the second is to improve readability. While using abbreviations is a quick way to save a few words, generally you will be better off spending more time (assuming the deadline is not pressing) thinking about why you have hit the word limit. Cutting words is often better done by saying things more concisely or leaving out unneeded details.
In regards to readability, the abstract is different than the body of the manuscript. While in both you need to consider readability by both experts and non-experts, the percentage of non-experts reading the abstract is higher than the percentage reading the body. Therefore, the readability of the abstract should be geared towards a non-expert. What this means is that while acronyms like ANN may help the expert reader, they will not help the non-expert. On the other hand, acronyms like NASA are more readable to the non-expert than National Aeronautics and Space Administration.
Abstract are tricky. For example, APA 5 style used to say that abstracts had to be self contained. That meant you had to introduce acronyms in the abstract and then again in the body. In APA 6, this has been dropped (cf. this blog post). I still go with introduce the acronym on first use in the abstract and then again on first use in the body. I also introduce acronyms on first use in figure captions (not sure that is APA style or Strongbad style).
Some journals’ policies do not let the author use abbreviations within the abstract, but they are not frequent, at least in my research realm.
Abbreviations are supposed to be invented and utilized to handle this very issue: repetition
It is worth mentioning that, in some cases, the abbreviated statement is well-known enough to the corresponding community. So, one could contend that the assertion of the expanded form might be redundant. As an instance, in the territory of robotic systems, the AI is always the famous representative of artificial intelligence and one does not explain about it, even within the abstract, typically. However, you better stick to the official rule, which says:
Introduce the abbreviation with its corresponding long phrase, once. Then, use the abbreviation, where ever you need.