## A Geometric Approach to Dynamical Model–Order Reduction

Any model order reduced dynamical system that evolves a modal decomposition to approximate the discretized solution of a stochastic PDE can be related to a vector field tangent to the manifold of fixed rank matrices. The Dynamically Orthogonal (DO) approximation is the canonical reduced order model for which the corresponding vector field is the orthogonal projection of the original system dynamics onto the tangent spaces of this manifold. The embedded geometry of the fixed rank matrix manifold is thoroughly analyzed. Geodesic equations are derived and extrinsic curvatures are characterized through the study of the Weingarten map. Differentiability results for the orthogonal projection onto embedded manifolds are reviewed and used to derive an explicit formula for the differential of the truncated Singular Value Decomposition (SVD). A similar analysis applied to the group of orthogonal matrices yields the differential of the polar decomposition. It is demonstrated that the error made by the DO approximation remains controlled under the minimal condition that the original solution stays close to the low rank manifold. Numerically, the DO approximation is also the dynamical system that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. The geometric analysis is used to provide improved numerical time-integration schemes. Riemannian matrix optimization including gradient and Newton methods allows to adaptively track the best low rank approximation of dynamical matrices.