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.

To make it very obvious as to what I'm trying to do, it's to essentially reverse this arrow here ^. Thank you.

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.

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.