Fact: One can easily compute heat dispersion in a plane using the heat equation.
 
Question: Has any research been done on computing the process in the reverse time direction? 

That is, given a heat map $\eta : \mathbb [0,1]^n \to \mathbb R$, can we find a $\nu : \mathbb [0,1]^n \to \mathbb R$ which heats over a 1-second period to satisfy $\int_{s \in [0,1]^n}|\nu^{\text{Heated for 1s}}(s) - \eta (s)| < \epsilon$ for small enough $\epsilon$. Say you can base $\epsilon$ on $|\eta^{\text{any # of derivatives}}| \le M$ and use $M$ as a variable, etc. Anything that makes the problem solvable or convenient to be solved and isn't trivial. 


If you just think about an infinitesimal in the forward time direction, $dt$, it comes natural that no information ought to be lost in this process, in theory. I did a search but I was unable to find any research papers or times where people have tried to compute heat transfer, say in the plane, in the reverse _time_ direction. 

[![enter image description here][1]][1]

To make it very obvious as to what I'm trying to do, it's to essentially reverse this arrow here ^. Thank you. Note that my plot is not showing all the information that is available, but that is simply an artifact of not showing enough subdivisions of the underlying space. 


  [1]: https://i.sstatic.net/Anh8o.png