What comes on top of a generator of a PDE?

[Omega0]

From some principles in nature we are using in physics the calculus of variations. Let me call it a generator for PDE's. My question: Are there levels above? What I mean is: Is there mathematics where you have principles where the solutions are generators for the generators for PDEs ?
What about generators for the generators for the generators?

 

[wrobel]

From some principles in nature we are using in physics the calculus of variations. Let me call it a generator for PDE's.

The heat equation for example does not come from a variational principle