# Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition

arXiv: https://arxiv.org/abs/1808.02086 · HTML: https://mlsanity.com/paper/1808.02086/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Aug 2018.

## Built on by (most cited first, 14 of 14)

- [Reduced operator inference for nonlinear partial differential equations](https://mlsanity.com/paper/2102.00083/) (2021)
- [Canonical and Noncanonical Hamiltonian Operator Inference](https://mlsanity.com/paper/2304.06262/) (2023)
- [A DeepONet multi-fidelity approach for residual learning in reduced order modeling](https://mlsanity.com/paper/2302.12682/) (2023)
- [Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference](https://mlsanity.com/paper/2308.13819/) (2023)
- [Context-aware controller inference for stabilizing dynamical systems from scarce data](https://mlsanity.com/paper/2207.11049/) (2022)
- [System stabilization with policy optimization on unstable latent manifolds](https://mlsanity.com/paper/2407.06418/) (2024)
- [Learning Latent Space Dynamics with Model-Form Uncertainties: A Stochastic Reduced-Order Modeling Approach](https://mlsanity.com/paper/2409.00220/) (2024)
- [Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method](https://mlsanity.com/paper/2501.07700/) (2025)
- [Nonlinear energy-preserving model reduction with lifting transformations that quadratize the energy](https://mlsanity.com/paper/2503.02273/) (2025)
- [Data-driven system identification using quadratic embeddings of nonlinear dynamics](https://mlsanity.com/paper/2501.08202/) (2025)
- [Stochastic Lifting for Generating Trajectories of Stochastic Physical Systems](https://mlsanity.com/paper/2605.29194/) (2026)
- [An adaptive data sampling strategy for stabilizing dynamical systems via controller inference](https://mlsanity.com/paper/2506.01816/) (2025)
- [Nonlinear Model Order Reduction of Dynamical Systems in Process Engineering: Review and Comparison](https://mlsanity.com/paper/2506.12819/) (2025)
- [Structure-preserving Lift & Learn: Scientific machine learning for nonlinear conservative partial differential equations](https://mlsanity.com/paper/2507.00301/) (2025)

## Abstract (arXiv)

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.