# Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators

arXiv: https://arxiv.org/abs/2106.09658 · HTML: https://mlsanity.com/paper/2106.09658/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 75 references that are in the map.
- First posted on arXiv 17 Jun 2021.

## Builds on (most cited first, 10 of 10)

- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Deep learning of thermodynamics-aware reduced-order models from data](https://mlsanity.com/paper/2007.03758/) (2020)
- [Machine Learning-augmented Predictive Modeling of Turbulent Separated Flows over Airfoils](https://mlsanity.com/paper/1608.03990/) (2016)
- [DR-RNN: A deep residual recurrent neural network for model reduction](https://mlsanity.com/paper/1709.00939/) (2017)
- [Long-time predictive modeling of nonlinear dynamical systems using neural networks](https://mlsanity.com/paper/1805.12547/) (2018)
- [Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics](https://mlsanity.com/paper/1401.8044/) (2014)
- [Dynamic mode decomposition for compressive system identification](https://mlsanity.com/paper/1710.07737/) (2017)

## Built on by (most cited first, 8 of 8)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification](https://mlsanity.com/paper/2204.12005/) (2022)
- [GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder](https://mlsanity.com/paper/2308.05882/) (2023)
- [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024)
- [Thermodynamically Consistent Machine-Learned Internal State Variable Approach for Data-Driven Modeling of Path-Dependent Materials](https://mlsanity.com/paper/2205.00578/) (2022)
- [Nonlinear proper orthogonal decomposition for convection-dominated flows](https://mlsanity.com/paper/2110.08295/) (2021)
- [Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEs](https://mlsanity.com/paper/2202.02658/) (2022)
- [FTL: Transfer Learning Nonlinear Plasma Dynamic Transitions in Low Dimensional Embeddings via Deep Neural Networks](https://mlsanity.com/paper/2404.17466/) (2024)

## Abstract (arXiv)

Although projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing such a reduced-order model typically requires significant modifications to the underlying simulation code. To address this, we propose a method that enables traditionally intrusive reduced-order models to be accurately approximated in a non-intrusive manner. Specifically, the approach approximates the low-dimensional operators associated with projection-based reduced-order models (ROMs) using modern machine-learning regression techniques. The only requirement of the simulation code is the ability to export the velocity given the state and parameters as this functionality is used to train the approximated low-dimensional operators. In addition to enabling nonintrusivity, we demonstrate that the approach also leads to very low computational complexity, achieving up to 1000× reduction in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs.