Optimal Galerkin approximations of partial differential equations using principal interaction patterns

A method of constructing minimal systems of ordinary differential equations modeling the dynamics of nonlinear partial differential equations is presented. Characteristic spatial structures called principal interaction patterns are extracted from the system according to a nonlinear variational principle based on a dynamical optimality criterion and used as basis functions in a Galerkin approximation. The potential of the method is illustrated using the Kuramoto-Sivashinsky equation as an example. As to the number of modes required to capture the dynamics of the complete system a reduced model based on principal interaction patterns yields a considerable improvement on more conventional approaches using Sobolev eigenfunctions or Karhunen-Lo`eve modes as basis functions and is far more efficient than a dynamical description based on Fourier modes.