Fig 1. Geometric illustration of flat and nonlinear Galerkin projected dynamics in ${\mathbb{R}}^3$.

3D manifold S living in (ξ₁, ξ₂, η₁) is projected to 2D plane parametrized by (ξ₁, ξ₂). Parametrization is assumed to be imperfect, i.e. out-of-plane coordinate η₁ cannot be uniquely determined from (ξ₁, ξ₂). Flat Galerkin method always uses the dynamics corresponding to η₁ = 0. Nonlinear Galerkin method uses the dynamics corresponding to η₁ = Φ(ξ₁, ξ₂) where Φ is determined by some prescribed criterion (e.g. minimization of L2 error).