# Learning Low-Dimensional Quadratic-Embeddings of High-Fidelity Nonlinear Dynamics using Deep Learning

arXiv: https://arxiv.org/abs/2111.12995 · HTML: https://mlsanity.com/paper/2111.12995/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 52 references that are in the map.
- First posted on arXiv 25 Nov 2021.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems](https://mlsanity.com/paper/1808.01346/) (2018)
- [Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms](https://mlsanity.com/paper/2002.09726/) (2020)
- [Operator Inference and Physics-Informed Learning of Low-Dimensional Models for Incompressible Flows](https://mlsanity.com/paper/2010.06701/) (2020)
- [Learning Dynamics from Noisy Measurements using Deep Learning with a Runge-Kutta Constraint](https://mlsanity.com/paper/2109.11446/) (2021)
- [On Reduced Input-Output Dynamic Mode Decomposition](https://mlsanity.com/paper/1712.08447/) (2017)
- [LQResNet: A Deep Neural Network Architecture for Learning Dynamic Processes](https://mlsanity.com/paper/2103.02249/) (2021)

## Built on by (most cited first, 2 of 2)

- [Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions](https://mlsanity.com/paper/2211.06786/) (2022)
- [Generalized Quadratic Embeddings for Nonlinear Dynamics using Deep Learning](https://mlsanity.com/paper/2211.00357/) (2022)

## Abstract (arXiv)

Learning dynamical models from data plays a vital role in engineering design, optimization, and predictions. Building models describing dynamics of complex processes (e.g., weather dynamics, or reactive flows) using empirical knowledge or first principles are onerous or infeasible. Moreover, these models are high-dimensional but spatially correlated. It is, however, observed that the dynamics of high-fidelity models often evolve in low-dimensional manifolds. Furthermore, it is also known that for sufficiently smooth vector fields defining the nonlinear dynamics, a quadratic model can describe it accurately in an appropriate coordinate system, conferring to the McCormick relaxation idea in nonconvex optimization. Here, we aim at finding a low-dimensional embedding of high-fidelity dynamical data, ensuring a simple quadratic model to explain its dynamics. To that aim, this work leverages deep learning to identify low-dimensional quadratic embeddings for high-fidelity dynamical systems. Precisely, we identify the embedding of data using an autoencoder to have the desired property of the embedding. We also embed a Runge-Kutta method to avoid the time-derivative computations, which is often a challenge. We illustrate the ability of the approach by a couple of examples, arising in describing flow dynamics and the oscillatory tubular reactor model.