When I looked at the heat equation I wondered if the solutions were always suppose to be like vector objects. What I rejected was that they are so smooth.
Then I remembered that you can maybe use recognition as a way to solve problems. So if we look at a poorly compressed image using JPEG compression we see that it is built up from non smooth connected blocks. The inside of those blocks are still smooth though.
Taking inspiration from this I can guess that you can probably get a better solution to certain problems solved by differential equations if you have multiple values for the same line around each smooth object.
That is. We surround each smooth object with an almost independent value line. This would create a problem with lots more of unknowns. To calculate or choose these we need some dependency information and that I guess would come from recognition some how. Maybe the smooth solution could be used as enough information.
I don’t know how to solve this. The idea is presented as a problem.