I am a postdoc who was supposed to co-supervise a PhD student along with another academic who invited me for that. However, I ended up fully supervising the student after the supervisor became too busy to attend meetings. Eventually, I also ended up defining a problem for the student, which, after 2 months of work, we discovered it is invalid. How should I deal with this situation? I feel that it's my fault and feel bad for the student. The student is in his first year, and this is his first research problem.
Any ideas on this from experienced supervisors over here? Thank you.
Edit: Thank you, everyone, for contributing your answers and comments. For those of you talking about failure/recovery/publishing what's in hand, I would clarify that the discovered issue is in the problem formulation, where the solution would involve inverting a rank-deficient matrix and differentiating a non-differentiable function. Thus, a mathematical solution is not defined. Therefore, the formulation is invalid. This wasn't foreseeable at the beginning, and only when the student made some deviations, it was possible to reach this conclusion. Now, in terms of publication and thesis content, the efforts exerted thus far don't mean anything.