Should I use abbreviations such as PDE or DN map in a paper?
If the term is sufficiently long and sufficiently frequently used, it is reasonable to use abbrevations, as it may be very helpful for reading. I once reviewed a paper that was almost unreadable due to using some very long and complicated term all over the place (and I recommended introducing an abbrevation). For example, compare the following:

Using a solver for partial differential equations to solve a problem after applying the Dirichlet-to-Neumann map was first suggested in Ref. 42.
Using a PDE solver for a DtN-mapped problem was first suggested in Ref. 42.

However, you should explain the abbrevation the first time you are using it, e.g.:

partial differential equation (PDE)

The only exception from this are abbrevations that are extremely common like JPEG or Laser, i.e., abbrevations you could also use in the title of a paper. Depending on your field, PDE might be such an abbrevation.
As a sidenote: I recommend to decide about the usage of abbrevations and explaining them in a piecewise manner. For example if you use a term as follows:

once in section 1,
not at all in section 2,
a lot in section 3,
not at all in section 4,
a lot in section 5 and 6.

I recommend not to use or introduce the abbrevation in section 1, but only use it in sections 3, 5 and 6, introducing it when it’s first used in section 3 and 5, respectively. Depending on how likely it is that somebody reads section 6 without reading section 5, it may also be wise to introduce it again at the beginning of section 6.