The AIC formula is :
$AIC = 2k + n Log(RSS/n)$
So if RSS is equal to 0, it is undefined. How do I deal with this? What value should it take?
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RSS = 0 implies that the model is a perfect fit since there was no residual. The limit of the log of 0 is $-\infty$, and since lower AICs are better, and this model is perfect,it makes perfect sense that the AIC should be a negative number such that no number can be lower.
As a follow-up to @Cameron Chandler; if your RSS=0 it should trigger some alerts since it is 100% overfitting, thus evaluating that model does not make any sense as such. But yes, take the limit of $\log(x), x\rightarrow 0^+$ and notice it goes towards $-\infty$ thus you cannot find a more perfekt model - if you only look at AIC, but you can (almost) not find a worse model if you look at the application. If you evaluate multiple fits I would simply just ignore those "perfect fits"