# Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems

arXiv: https://arxiv.org/abs/1912.08177 · HTML: https://mlsanity.com/paper/1912.08177/

## Key facts

- Cited by 96 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Dec 2019.

## Built on by (most cited first, 30 of 96)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
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- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
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- [SVD Perspectives for Augmenting DeepONet Flexibility and Interpretability](https://mlsanity.com/paper/2204.12670/) (2022)
- [One-shot learning for solution operators of partial differential equations](https://mlsanity.com/paper/2104.05512/) (2021)
- [gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification](https://mlsanity.com/paper/2204.12005/) (2022)
- [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022)
- [An Extensible Benchmark Suite for Learning to Simulate Physical Systems](https://mlsanity.com/paper/2108.07799/) (2021)
- [Generative learning for nonlinear dynamics](https://mlsanity.com/paper/2311.04128/) (2023)
- [Reduced operator inference for nonlinear partial differential equations](https://mlsanity.com/paper/2102.00083/) (2021)
- [Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2303.17078/) (2023)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Multi-fidelity reduced-order surrogate modeling](https://mlsanity.com/paper/2309.00325/) (2023)
- [Learning to Forecast Dynamical Systems from Streaming Data](https://mlsanity.com/paper/2109.09703/) (2021)
- [Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference](https://mlsanity.com/paper/2107.02597/) (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)
- [CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equations](https://mlsanity.com/paper/2402.14646/) (2024)
- [Physics-informed active learning with simultaneous weak-form latent space dynamics identification](https://mlsanity.com/paper/2407.00337/) (2024)
- [Operator inference with roll outs for learning reduced models from scarce and low-quality data](https://mlsanity.com/paper/2212.01418/) (2022)
- [Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference](https://mlsanity.com/paper/2308.13819/) (2023)
- [PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator](https://mlsanity.com/paper/2306.12962/) (2023)
- [Training multi-objective/multi-task collocation physics-informed neural network with student/teachers transfer learnings](https://mlsanity.com/paper/2107.11496/) (2021)
- [Generalized Quadratic Embeddings for Nonlinear Dynamics using Deep Learning](https://mlsanity.com/paper/2211.00357/) (2022)

## Abstract (arXiv)

We present Lift & Learn, a physics-informed method for learning low-dimensional models for large-scale dynamical systems. The method exploits knowledge of a system's governing equations to identify a coordinate transformation in which the system dynamics have quadratic structure. This transformation is called a lifting map because it often adds auxiliary variables to the system state. The lifting map is applied to data obtained by evaluating a model for the original nonlinear system. This lifted data is projected onto its leading principal components, and low-dimensional linear and quadratic matrix operators are fit to the lifted reduced data using a least-squares operator inference procedure. Analysis of our method shows that the Lift & Learn models are able to capture the system physics in the lifted coordinates at least as accurately as traditional intrusive model reduction approaches. This preservation of system physics makes the Lift & Learn models robust to changes in inputs. Numerical experiments on the FitzHugh-Nagumo neuron activation model and the compressible Euler equations demonstrate the generalizability of our model.