Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition

• Published in: AIAA journal 2019-06, Vol.57 (6), p.2297-2307
• Main Authors: Kramer, Boris, Willcox, Karen E
• Format: Article
• Language: English
• Subjects: Differential equations; Empirical analysis; Hoisting; Interpolation; Mathematical models; Model accuracy; Model reduction; Ordinary differential equations; Polynomials; Proper Orthogonal Decomposition; Reduced order models; Transformations (mathematics)
• ISSN: 0001-1452; 1533-385X
• DOI: 10.2514/1.J057791

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 precomputed. Thus, a key benefit of the approach is that there is no need for additional approximations of nonlinear terms, which is 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.