  In this paper we present a continuation method which transforms spatially
distributed ODE systems into continuous PDE. We show that this continuation can
be performed both for linear and nonlinear systems, including multidimensional,
space- and time-varying systems. When applied to a large-scale network, the
continuation provides a PDE describing evolution of continuous state
approximation that respects the spatial structure of the original ODE. Our
method is illustrated by multiple examples including transport equations,
Kuramoto equations and heat diffusion equations. As a main example, we perform
the continuation of a Newtonian system of interacting particles and obtain the
Euler equations for compressible fluids, thereby providing an original
alternative solution to Hilbert's 6th problem. Finally, we leverage our
derivation of the Euler equations to control multiagent systems, designing a
nonlinear control algorithm for robot formation based on its continuous
approximation.
