You can't automatically replace the Magic Initiate spells later on
If a rule allows for retraining it would say so. Magic Initiate does not have any such rule.
Continued study would be simulated by Multiclassing
Whatever character your playing is progressing as a main class, and most of their attention is dedicated to that progression. The Magic Initiate feat captures a small amount of attention devoted to another class, but continued study (such that you would acquire new spells) would be emblematic of a multiclass wizard.
(this section is pending revision upon further debate here)
If you gain a level in a class that has Known Spells for which you also have your Magic Initiate spells, you could use the Spellcasting feature to replace it:
Additionally, when you gain a level in this class, you can choose one of the [class] spells you know and replace it with another spell from the [class] spell list, which also must be of a level for which you have spell slots.
These spells still count as [class] spells since they appear on that class's spell list. As such, it is eligible for replacement by the Spellcasting feature, but will still be subject to the rules that:
Once you cast it, you must finish a long rest before you can cast it again using this feat.