  Stochastic dynamical systems with continuous symmetries arise commonly in
nature and often give rise to coherent spatio-temporal patterns. However,
because of their random locations, these patterns are not well captured by
current order reduction techniques and a large number of modes is typically
necessary for an accurate solution. In this work, we introduce a new
methodology for efficient order reduction of such systems by combining (i) the
method of slices, a symmetry reduction tool, with (ii) any standard order
reduction technique, resulting in efficient mixed symmetry-dimensionality
reduction schemes. In particular, using the Dynamically Orthogonal (DO)
equations in the second step, we obtain a novel nonlinear Symmetry-reduced
Dynamically Orthogonal (SDO) scheme. We demonstrate the performance of the SDO
scheme on stochastic solutions of the 1D Korteweg-de Vries and 2D Navier-Stokes
equations.
