  Consider the generalized Kuramoto-Sivashinsky (gKS) equation. It is a model
prototype for a wide variety of physical systems, from flame-front propagation,
and more general front propagation in reaction-diffusion systems, to interface
motion of viscous film flows. Our aim is to develop a systematic and rigorous
low-dimensional representation of the gKS equation. For this purpose, we
approximate it by a renormalization group (RG) equation which is qualitatively
characterized by rigorous error bounds. This formulation allows for a new
stochastic mode reduction guaranteeing optimality in the sense of maximal
information entropy. Herewith, noise is systematically added to the reduced gKS
equation and gives a rigorous and analytical explanation for its origin. These
new results would allow to reliably perform low-dimensional numerical
computations by accounting for the neglected degrees of freedom in a systematic
way. Moreover, the presented reduction strategy might also be useful in other
applications where classical mode reduction approaches fail or are too
complicated to be implemented.
