# A Scalable Observation-Driven Time-Dependent Basis for a Reduced Description of Transient Systems

## TL;DR

This paper introduces a scalable, observation-driven variational method to extract a time-dependent basis for transient systems, enabling reduced-order modeling without requiring knowledge of the underlying system model.

## Contribution

It presents a novel variational principle-based approach for deriving a time-dependent orthonormal basis solely from system observations, applicable to unknown systems.

## Key findings

- Method effectively captures transient dynamics in diverse systems
- Outperforms static basis methods like POD and DMD in transient scenarios
- Computationally efficient with linear complexity and memory requirements

## Abstract

We present a variational principle for the extraction of a time-dependent orthonormal basis from random realizations of transient systems. The optimality condition of the variational principle leads to a closed-form evolution equation for the orthonormal basis and its coefficients. The extracted modes are associated with the most transient subspace of the system, and they provide a reduced description of the transient dynamics that may be used for reduced-order modeling, filtering and prediction. The presented method is matrix-free and relies only on the observables of the system and ignores any information about the underlying system. In that sense, the presented reduction is purely observation-driven and may be applied to systems whose models are not known. The presented method has linear computational complexity and memory storage requirement with respect to the number of observables and the number of random realizations. Therefore, it may be used for a large number of observations and samples. The effectiveness of the proposed method is tested on three examples: (i) stochastic advection equation, (ii) a reduced description of transient instability of Kuramoto-Sivashinsky, and (iii) a transient vertical jet governed by incompressible Navier-Stokes equation. In these examples, we contrast the performance of the time-dependent basis versus static basis such as proper orthogonal decomposition, dynamic mode decomposition and polynomial chaos expansion.