Tensorial parametric model order reduction of nonlinear dynamical systems

TL;DR

This paper introduces a tensorial reduced-order modeling approach for nonlinear parametric dynamical systems, leveraging low-rank tensor approximations to improve efficiency, accuracy, and generalization to unseen parameters.

Contribution

It develops a novel tensor-based ROM framework that incorporates parameter dependence using multilinear algebra, enhancing prediction for unseen parameters and reducing computational costs.

Findings

Tensorial ROM predicts unseen parameter dynamics effectively.

Reduces online computational costs independent of full model size.

Achieves lower reduced dimensions compared to traditional methods.

Abstract

For a nonlinear dynamical system that depends on parameters, the paper introduces a novel tensorial reduced-order model (TROM). The reduced model is projection-based, and for systems with no parameters involved, it resembles proper orthogonal decomposition (POD) combined with the discrete empirical interpolation method (DEIM). For parametric systems, TROM employs low-rank tensor approximations in place of truncated SVD, a key dimension-reduction technique in POD with DEIM. Three popular low-rank tensor compression formats are considered for this purpose: canonical polyadic, Tucker, and tensor train. The use of multilinear algebra tools allows the incorporation of information about the parameter dependence of the system into the reduced model and leads to a POD-DEIM type ROM that (i) is parameter-specific (localized) and predicts the system dynamics for out-of-training set (unseen) parameter values, (ii) mitigates the adverse effects of high parameter space dimension, (iii) has online computational costs that depend only on tensor compression ranks but not on the full-order model size, and (iv) achieves lower reduced space dimensions compared to the conventional POD-DEIM ROM. The paper explains the method, analyzes its prediction power, and assesses its performance for two specific parameter-dependent nonlinear dynamical systems.