# Model order reduction for stochastic dynamical systems with continuous symmetries

## TL;DR

This paper introduces a new order reduction method for stochastic dynamical systems with continuous symmetries, combining symmetry reduction with standard techniques to efficiently capture coherent patterns.

## Contribution

The authors develop a novel symmetry-reduced Dynamically Orthogonal (SDO) scheme that improves efficiency in modeling stochastic systems with symmetries.

## Key findings

- SDO scheme effectively captures patterns with fewer modes
- Demonstrated on 1D Korteweg-de Vries and 2D Navier-Stokes equations
- Outperforms traditional methods in symmetry-invariant stochastic systems

## Abstract

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.