How to solve this regression problem?
If you want to predict, you need an underlying model of the behavior. Then you can perform least-squares fitting to determine the model parameters. Linear regression just assumes that you have a linear relationship and predicts from that relationship - there's nothing magic about it, except that the model is simple and often effective. Polynomial regression gives bad behavior when extrapolating, so I wouldn't advise that.

Building a model function is usually a trial-and-error process (unless you have some prior knowledge about your system, which I guess you don't). I'd suggest looking at the locus of (x, y) points where z is maximized. It looks like they have an exponential decay relationship. A model like y = exp(-x/x0) might fit the data rather closely. Once you've determined the value of x0 from the data, you can rebuild a model for z.