Replacing large-dimensional ODE systems with one PDE Is it possible to replace a large-dimensional system of differential equations with one partial differential equation?
 A: Let me single out a situation which goes the other way around: how a system of ODE is describing the propagation of singularities for a principal type PDE. 
Take a linear (pseudo)differential operator of real principal type with smooth coefficients: the principal symbol $p(x,\xi)$ is real-valued and $dp\wedge \xi\cdot dx\not=0$ (verified for the wave equation or  a non-vanishing real vector field). Then the singularities are moving along the bicharacteristic curves, which are the integral curves of the Hamiltonian vector field of $p$,
$$
H_p=\frac{\partial p}{\partial \xi}\cdot \frac{\partial }{\partial x}-
\frac{\partial p}{\partial x}\frac{\partial }{\partial \xi}.\quad 
$$
Solving the system of ODE, $\dot \Gamma=H_p(\Gamma)$ is enough to understand the propagation of singularities: if $p(x,D) u\in C^\infty$ the the wave-front-set of $u$ is invariant by the flow of the Hamiltonian vector field. There is no need to solve the PDE if you are only interested in singularities.
A: This idea is studied in the field "kinetic PDE." See these lecture notes by Clément Mouhot:
http://cmouhot.wordpress.com/1900/10/25/mathematical-topics-in-kinetic-theory-part-iii-course/
Ch 2 in particular should be of interest.
