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you may recall (okay, probably not ;-) what I once wrote in the entry on exterior differential systems: while in the classical literature these are thought of as dg-ideals in a de Rham complex, we should think of them as sub-Lie algebroids of tangent Lie algebroids.
Since exterior differential systems over X encode and are encoded by partial differential equations on functions on X, this means that such sub-Lie algebroids are partial differential equations.
This perspective is amplified much more in the literature on D-modules: I think we can think of a D-scheme as an infinite-order analog of a Lie algebroid, which is the corresponding first-order notion. The Jet-bundle with its D-scheme structure is the infinite-order analog of the tangent Lie algebroid.
And sub-D-schemes of Jet-D-schemes are partial differential equations, this is what everyone on D-geometry tells you first.
So I think there is a nice story here.
In a viewpoint, D-modules are based on $D_X$ of course, the Grothendieck’s notion of a regular differential operator. The PDE people look more generally, allowing diff operators more general coefficients in front, and allowing more general, weak solutions of various kinds and the main business is that sometimes the apparent generality reduces a lot for good equations and one has increase of regularity for free. Another interesting direction, which is very relevant for physics, and was discovered in mathematical physics by Maslov independently is the calculus of pseudodifferential operators and more, generally, Fourier differential operators. Part of the effort in the microlocal analysis of Japanese school of Michio Sato and Masaki Kashiwara was to extends the D-module methods to such setups.
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