Just thought I'd mention that Einstein himself proposed an interesting method for "counting" the solutions of a PDE which may have been indirectly inspired by the landmark work by his colleague David Hilbert on what is now called "the hilbert polynomial" (both ideas involve analyzing the asymptotics of dimension counting in a series of finite dimensional vector spaces). See http://en.wikipedia.org/w/index.php?title=Constraint_counting&oldid=39565830 (notice that I cited a specific version, namely the one listed at http://en.wikipedia.org/wiki/User:Hillman/Archive, bearing in mind that Wikipedia articles can change drastically in seconds and that I haven't checked the current version, just the last version I worked on myself.) For example, by counting alone we can conclude that a solution to the standard one dimensional wave equation should be specified by two freely chosen real functions of one real variable each, and of course just such a representation is known.

His charming idea has been taken up in the context of gtr by Siklos and (independently) Schutz, and others; see the 1996 paper by Siklos cited in the above cited Wikipedia article, which is one of my favorites. Siklos shows that from counting alone, we should expect a vacuum solution to be given by four real functions of three real variables, plus six real functions of two real variables, and in fact such a representation is known.

See also the more recent paper by Llosa and Soler, CQG 22 (2005).