Current noise-cancelling headphones deal well with predictable noise that has a short period of repetition, for example engine rumbling or the hum of an air-conditioner or fan. Unpredictable noise is of course difficult to cancel – the headphones would have to detect the new sound and produce the opposite wave of air pressure quicker than the human ear can detect the new sound. This is theoretically possible, because the sound reaches the outside of the headphone before it reaches the inside of the ear, but may not be feasible at the current technology level.
What is possible, but not done, at least by the Sony MDR-1000x headphones I have, is cancelling predictable noise with a longer period of repetition. Specifically, the beeping sound of trucks reversing has a period of 1-2s and is very predictable, but the headphones do not cancel it at all. It seems that a tweak of the noise-prediction algorithm could fix this – no need to invoke machine learning or anything more complicated. The headphones would just have to keep track of the sounds reaching them in the last few seconds and look for simple repeating patterns. Then these patterns can be predicted and cancelled. Currently the headphones seem to predict only based on the last half-second or less, so any longer repetitions of sound are not taken into account.
Some birdsong is repeated beeping, similar to the signal of trucks reversing, but of course slightly less predictable. This bird-noise could conceivably also be cancelled, although if the gaps between the beeps vary, then the first small length of time during an unexpectedly early beep would be difficult. Similarly, if the length of the beeps varies, then a beep that stops unexpectedly early would be over-cancelled (headphones produce a sound that is detectable on the background of silence).
To help the headphones recognise new noise patterns, the user can press a button when an undesirable sound is heard, and release the button when the sound stops. The algorithm can compare the button presses to its sound-recording in the same time interval, which would help it identify the start and end of the noise that needs to be cancelled. Sometimes humans are better at detecting complex patterns than a computer, in which case this user input to the headphones would speed up the identification of new forms of noise.