Is there an analogue of coherence between data sets related by a non-linear transformation? Calculating the coherence (sometimes called magnitude squared coherence) between two signals indicates the presence or lack thereof of a linear transformation between the two signals.  Is there an analogue for signals which are related by a non-linear transformation?  Let's assume that the expected non-linear transformation is known.  It would be even better if the calculation could identify the non-linear transformation within some bounds, but that is probably just wishful thinking.

An example of coherence between ocean water level (due to tides) and ground water level measured in a well, taken from the Wikipedia article linked above.  This is an example of the coherence between two signals related by a linear transformation.
Are you interested in coherence's ability to discriminate correlations at different frequencies, or are you interested in any measure of non-linear correlation? Coherence is basically cross-correlation in the frequency domain.

Mutual information can measure non-linear correlations, but says nothing about the form of the correlation. It's also difficult to estimate accurately without large amounts of data (kernel density estimation is a popular tool to help with that right now, though I think cupolas have begun to gain traction).

I don't know that anybody has done so, but you could alternatively use one signal to predict another using an artificial neural network with one hidden layer. Your correlation metric could then be something like an RMSE between predicted and target signals. It feels like a hack, though. With some work you could try to infer something about the relationship type based on the trained weights of the network. I'm not an expert, but I don't know of a formal framework to actually do that. It's also not an established metric, so you have to also do the work to convince people that it's meaningful (much more difficult if you can't cite sources that did it before you).