Computer analysis predicted rises, ebbs in Afghanistan violence
What Asimov was talking about was taking large groups of humans who were unaware of the predictions and modeling the whole sweep of society analogous to the kinetic theory of gases
where individual items are just that - individual and impossible to predict - but large masses develop a sort of bulk tendency. A more modern theory would have to account for aspects of chaos theory
. The article I link to seems to be using large amounts of data and sort of brute-force calculating a specific type of behavior in a specific place. Whatever the specifics, something like psychohistory is probably possible and aspects of it are becoming more and more apparent, such as the above study (if valid). But whether it's possible to actually "predict the future", I don't know. It would necessarily have to be a statistical thing - the weatherman said it was going to rain today and it didn't. A psychohistorian might say it's highly likely that we'll have a war or a recession or whatever, and I don't think it could ever be 100%. It does raise interesting philosophical questions of freewill and determinism and complicates it by putting it on a species/social level. But, in terms of whether it's "a thing", I'd say something analogous to it is a small but growing thing. The more data and the more computational power, the more you can produce predictions that aren't wild guesses. So the more feasible it becomes to test those predictions and refine the principles you use to make them. So the more we'll find out whether it, or something like it, can indeed be a science.
But I'm not at all versed in the scientific literature on the subject - it could be here and I wouldn't know it (and, indeed, the corporations and governments with the computational power might not broadcast it) or it could be completely infeasible and I just haven't come across the reason why.