Here's a summary experiment where a computer can predict a subject's decisions seconds before the subject is conscious of the decision. This experiment seems to make observations on the relationship between physical biological processes and the ultimate subjective, conscious, perception of "having made a decision".
Usually, these experiments are done using real-time f-MRI measurements to detect brain state, and then associating different MRI data patterns with different mental states.
To me, the main issue with these is in identifying the time of "conscious decision" making, to address this different labs have tried to set up different protocols to tease out the exact time of "conscious" decision making, e.g.
M. Matsuhashi and M. Hallett "The timing of the conscious intention to move", European Journal of Neuroscience 2008 used the idea of when a subject was able to respond to an external stimulus to veto an action.
Obligatory Wikipedia article
I think that, in net, these experiments do represent progress towards understanding the relationship between brain states and mental states, and have significant philosophical implications.
Note: the first linked article is a popularization, but includes a reference to the qualified scientific research.