# Model order reduction for random nonlinear dynamical systems and low-dimensional representations for their quantities of interest

## TL;DR

This paper explores model order reduction techniques, specifically proper orthogonal decomposition, for efficiently approximating quantities of interest in random nonlinear dynamical systems affected by uncertainty.

## Contribution

It introduces a projection-based model reduction approach for large stochastic dynamical systems, providing error analysis and numerical validation.

## Key findings

- Reduced-order models significantly decrease computational complexity.
- Error bounds for low-dimensional approximations are established.
- Numerical examples demonstrate effectiveness of the method.

## Abstract

We examine nonlinear dynamical systems of ordinary differential equations or differential algebraic equations. In an uncertainty quantification, physical parameters are replaced by random variables. The inner variables as well as a quantity of interest are expanded into series with orthogonal basis functions like the polynomial chaos expansions, for example. On the one hand, the stochastic Galerkin method yields a large coupled dynamical system. On the other hand, a stochastic collocation method, which uses a quadrature rule or a sampling scheme, can be written in the form of a large weakly coupled dynamical system. We apply projection-based methods of nonlinear model order reduction to the large systems. A reduced-order model implies a low-dimensional representation of the quantity of interest. We focus on model order reduction by proper orthogonal decomposition. The error of a best approximation located in a low-dimensional subspace is analysed. We illustrate results of numerical computations for test examples.