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Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition

Boris Kramer, Karen Willcox

arXiv:1808.02086v2math.NAcs.NAcs.SYeess.SYmath.DS

TL;DR

Nonlinear model reduction lacks the mature theoretical understanding available for linear systems, while general nonlinear POD can require careful tuning. This paper lifts nonlinear models into structured polynomial systems and applies POD; the resulting ROMs are reported as competitive with state-of-the-art POD-DEIM methods, with potential advantages for analysis and input-independent modeling.

  • Problem

    For general nonlinear systems, POD often depends on careful tuning, and rigorous theory is less developed than for linear systems.

    Source: §1 · p. 2

  • Method

    The method lifts a nonlinear system into an equivalent polynomial system with auxiliary variables, then applies POD while preserving the lifted structure.

    Source: §1 · p. 3, §3 · p. 9

  • Results

    The lifted ROMs are competitive with state-of-the-art nonlinear model reduction methods in the reported examples.

    Source: §1 · p. 3, §6 · p. 21

  • Takeaways & Limitations

    The structured polynomial models are presented as promising for rigorous ROM analysis and input-independent reduced-order modeling.

    Source: §1 · p. 3

    Projecting the lifted algebraic constraints or auxiliary dynamics onto a reduced subspace also approximates the original nonlinearity.

    Source: §5.4 · p. 18

Abstract

from arXiv · show

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.

1 Introduction

The paper addresses the gap between well-developed theory for linear model reduction and the challenges of nonlinear systems, proposing lifting to give general nonlinear models a polynomial structure. It combines lifted-system POD with structure-preserving reduction and reports accuracy competitive with state-of-the-art methods, alongside potential benefits for analysis and input-independent modeling.

1 Introduction

  • Lifting transforms general nonlinear systems into equivalent polynomial systems by introducing auxiliary variables and variable transformations.The paper targets polynomial structure, including quadratic-bilinear systems, for POD-based reduction.Source: §1 · pp. 2–3
  • Polynomial-system ROMs allow reduced-order operators to be precomputed without sampling-based approximation of nonlinear functions.The introduction contrasts this property with general nonlinear ROMs that need an additional approximation step.Source: §1 · p. 3
  • The structured lifted models are presented as promising for rigorous ROM analysis and input-independent reduced-order modeling.The introduction names stability and error analysis as examples of properties that structured systems may help analyze.Source: §1 · p. 3
  • Earlier lifting work targeted QB-ODEs, whereas this paper’s multi-step transformations also yield QB-DAEs or quartic ODEs.The introduction also situates variable transformations across optimization, differential equations [24, 37], control, and Koopman analysis [43].Source: §1 · pp. 2–3
  • The lifted POD models are competitive with state-of-the-art nonlinear model reduction, while retaining polynomial or quadratic structure.The introduction identifies the tubular reactor as the strongly nonlinear example and describes two lifted forms: a quartic ODE and a QB-DAE.Source: §1 · p. 3

2 Polynomial Systems and Proper Orthogonal Decomposition Model Reduction

This section develops POD reduction for polynomial systems, relating the full-order cost of general nonlinear POD to polynomial models whose reduced operators can be precomputed. For QB-DAEs, the projection preserves the DAE index and algebraic structure.

2.1 Proper Orthogonal Decomposition Model Reduction

  • Standard POD builds a basis from solution snapshots and Galerkin-projects the model, but evaluating the projected nonlinear term still scales with full-order dimension.Hyper-reduction can require many sampling points for strong nonlinearities and adds an approximation that can hinder stability and error analysis.Source: §2.1 · pp. 5–6

2.2 Polynomial Systems and Proper Orthogonal Decomposition

  • Lifting can convert general nonlinear systems to polynomial form; for quartic and quadratic systems, POD yields reduced operators by direct projection that can all be precomputed.This polynomial POD approach avoids an extra hyper-reduction step, though ROM quality can depend on training data, selected modes, and system properties.Source: §2.2 · pp. 6–7

2.3 Quadratic-Bilinear Systems and Proper Orthogonal Decomposition

  • POD projection of a structured QB-DAE preserves its DAE index and algebraic equations, and its reduced operators can be precomputed without DEIM or other hyper-reduction.The model separates dynamically evolving states from algebraically constrained variables.Source: §2.3 · pp. 8–9

3 Lifting Transformations

Lifting transforms a nonlinear system into a larger, equivalent system with auxiliary variables and more structure, targeting a polynomial form suitable for POD. Allowing algebraic constraints can keep the lifted system smaller, but representations are nonunique and may affect system properties.

3 Lifting Transformations

  • Lifting adds auxiliary variables to form a larger, equivalent system with more structure, targeting a polynomial representation amenable to POD.The transformation introduces no approximation.Source: §3 · pp. 9–11
  • In the example, introducing successive auxiliary states turns the original nonlinear equation into an equivalent quadratic-bilinear ODE.An alternative two-variable QB-DAE retains the algebraic constraint w1 = x^2; both representations yield the same x(t).Source: §3 · pp. 9–10
  • Allowing differential-algebraic equations can keep the number of auxiliary variables low, even when lifting to a quadratic-bilinear ODE is possible.The paper notes that this matters for discretized PDEs, where the original state dimension is already large.Source: §3 · p. 10
  • Equivalent polynomial lifts are nonunique, and the authors know of no algorithm for finding a minimal one.Different lifting choices can also influence differential-equation stiffness.Source: §3 · p. 10

4 Benchmark Problem: FitzHugh-Nagumo

The FitzHugh-Nagumo benchmark tests lifting and POD-based reduction on a spiking-neuron model. The lifted QB-POD model's error decreases with added basis functions, unlike POD-DEIM errors that flatten, while its reduced trajectories capture the full model's limit-cycle oscillations.

4 Benchmark Problem: FitzHugh-Nagumo

  • The benchmark models activation and deactivation of a spiking neuron and has been studied with DEIM, balanced reduction, and interpolation-based reduction.The cited prior approaches are Ref., Ref. [6], and Ref., respectively.Source: §4 · p. 11

4.2 FitzHugh-Nagumo Lifted Formulation

  • Setting z = v^2 makes the FitzHugh-Nagumo equations quadratic; the auxiliary variable requires consistent initial conditions and chain-rule boundary conditions.Finite differences discretize each of the three variables with n = 512 degrees of freedom, yielding a QB-ODE of dimension 3n = 1536.Source: §4.2 · p. 12

4.3 FitzHugh-Nagumo Lifted Quadratic-Bilinear Reduced-Order Model

  • POD bases use the first 100 of 150 snapshots through t = 8s, so the reduced models predict 50% beyond training data.The auxiliary snapshot z = v^2 has the same singular-value decay as v, up to numerical accuracy.Source: §4.3 · pp. 12–13
  • With 3r = 9 reduced states, QB-POD captures the full model's limit-cycle oscillations and visually matches its quantities of interest.The comparison uses w(0, t) and v(0, t).Source: §4.3 · p. 13, Figure 2 · p. 13
  • Lifted QB-POD error steadily decreases with ROM dimension, whereas POD-DEIM errors flatten; the lifted approach recovers regular POD accuracy without extra nonlinear-term approximation.Figure 3 compares average relative state error against overall ROM dimension.Source: Figure 3 · p. 14, §4.3 · p. 14

5 Application: Tubular Reactor Model

The tubular reactor’s exponential nonlinearity is lifted into equivalent polynomial formulations and reduced with POD. The resulting ROMs reproduce reactor behavior and compare favorably with POD-DEIM, with accuracy affected by model regime and basis choices.

5.1 Partial Differential Equation Model and Discretization

  • The discretized reactor model has 2n states and an exponential nonlinearity, so direct POD requires an additional approximation such as DEIM.Source: §5.1 · pp. 15–16

5.2 Lifted Model 1: A Quartic Ordinary Differential Equation

  • With consistent auxiliary initial conditions, this fourth-order polynomial ODE has identical ψ and θ solutions to original ODEs (27)–(28), while discretization dimension increases from 2n to 5n.Source: §5.2 · p. 16

5.3 Lifted Model 2: A Quadratic-Bilinear Differential-Algebraic System

  • Further lifting to quadratic-bilinear form requires algebraic constraints, since further differentiating the new variables would not yield a QB system without more auxiliaries.Source: §5.3 · pp. 16–17

5.4 Proper Orthogonal Decomposition for Quartic and Quadratic-Bilinear Differential-Algebraic Equations

  • POD uses separate snapshot-based bases for each lifted variable; projecting constrained states into a subspace approximates the lifting relations, although hyper-reduction is unnecessary.Source: §5.4 · pp. 17–18

5.5 Numerical Results

  • Snapshots stop at t=20 s while models predict to 30 s; singular values decay faster in the stable case, while w1, w4, and w6 decay more slowly in the unstable case.Source: §5.5 · pp. 18–19, Figure 4 · p. 19
  • With r1=30 and r2=9, the QB-DAE ROM reproduces stable and limit-cycle reactor-exit temperature oscillations; average relative state errors are 6.71 × 10−5 stable and 8.95 × 10−3 unstable.Source: §5.5 · pp. 19–20, Figure 5 · p. 20
  • Across Figure 6’s comparisons, fixed-count DEIM stops improving as modes are added, while the quartic ROM improves and QB-DAE accuracy rises with a better constrained-state basis.Source: §5.5 · pp. 20–21, Figure 6 · p. 21

6 Conclusions

The paper finds lifting with structure-preserving POD competitive with POD-DEIM on the chosen examples. It also identifies costs and possible directions for analysis and projection.

  • For the chosen examples, lifting with structure-preserving POD is competitive in reduced-model accuracy with POD-DEIM.Source: §6 · p. 21
  • The approach avoids additional approximation of reduced nonlinear terms but requires upfront work to derive the lifted system and its matrices and tensors.Source: §6 · p. 21
  • Auxiliary variables increase state and basis dimensions; known lifting transformations may support nonlinear projection and more rigorous ROM stability and error analysis.Source: §6 · p. 21

Appendix: Details on Solving the QB-DAE ROM

The appendix describes ways to solve the QB-DAE ROM and precompute tensor products for faster online simulation.

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