Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition

TL;DR

This paper introduces a nonlinear model reduction technique that uses variable transformations to lift the system into a structured polynomial form, enabling efficient POD-based reduction without additional nonlinear approximations.

Contribution

The method lifts nonlinear systems into a polynomial form through variable transformations, allowing POD reduction without hyper-reduction or sparse sampling, and facilitates rigorous analysis.

Findings

Competitive accuracy with state-of-the-art methods

No need for hyper-reduction or sparse sampling

Applicable to systems with general nonlinearities

Abstract

This paper presents a structure-exploiting nonlinear model reduction method for systems with general nonlinearities. First, the nonlinear model is lifted to a model with more structure via variable transformations and the introduction of auxiliary variables. The lifted model is equivalent to the original model---it uses a change of variables, but introduces no approximations. When discretized, the lifted model yields a polynomial system of either ordinary differential equations or differential algebraic equations, depending on the problem and lifting transformation. Proper orthogonal decomposition (POD) is applied to the lifted models, yielding a reduced-order model for which all reduced-order operators can be pre-computed. Thus, a key benefit of the approach is that there is no need for additional approximations of nonlinear terms, in contrast with existing nonlinear model reduction methods requiring sparse sampling or hyper-reduction. Application of the lifting and POD model reduction to the FitzHugh-Nagumo benchmark problem and to a tubular reactor model with Arrhenius reaction terms shows that the approach is competitive in terms of reduced model accuracy with state-of-the-art model reduction via POD and discrete empirical interpolation, while having the added benefits of opening new pathways for rigorous analysis and input-independent model reduction via the introduction of the lifted problem structure.