  Reduced-order models of time-dependent partial differential equations (PDEs)
where the solution is assumed as a linear combination of prescribed modes are
rooted in a well-developed theory. However, more general models where the
reduced solutions depend nonlinearly on time varying parameters have thus far
been derived in an ad hoc manner. Here, we introduce Reduced-order Nonlinear
Solutions (RONS): a unified framework for deriving reduced-order models that
depend nonlinearly on a set of time-dependent parameters. The set of all
possible reduced-order solutions are viewed as a manifold immersed in the
function space of the PDE. The parameters are evolved such that the
instantaneous discrepancy between reduced dynamics and the full PDE dynamics is
minimized. This results in a set of explicit ordinary differential equations on
the tangent bundle of the manifold. In the special case of linear parameter
dependence, our reduced equations coincide with the standard Galerkin
projection. Furthermore, any number of conserved quantities of the PDE can
readily be enforced in our framework. Since RONS does not assume an underlying
variational formulation for the PDE, it is applicable to a broad class of
problems. We demonstrate the efficacy of RONS on three examples: an
advection-diffusion equation, the nonlinear Schrodinger equation and Euler's
equation for ideal fluids.
