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

@inproceedings{Goyal2021LearningLQ, title={Learning Low-Dimensional Quadratic-Embeddings of High-Fidelity Nonlinear Dynamics using Deep Learning}, author={Pawan Goyal and Peter Benner}, year={2021} }

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…

## Figures from this paper

## 2 Citations

### Operator inference for non-intrusive model reduction with nonlinear manifolds

- Mathematics, Computer ScienceArXiv
- 2022

This paper proposes a novel approach for learning a data-driven quadratic manifold from high-dimensional data, then employing the quadRatic manifold to derive eﬃcient physics-based reduced-order models, using a polynomial mapping between high- dimensional states and a low-dimensional embedding.

### Operator inference for non-intrusive model reduction with quadratic manifolds

- Computer Science, Mathematics
- 2022

This paper proposes a novel approach for learning a data-driven quadratic manifold from high-dimensional data, then employing this quadratic manifold to derive eﬃcient physics-based reduced-order…

## References

SHOWING 1-10 OF 51 REFERENCES

### Deep learning for universal linear embeddings of nonlinear dynamics

- Computer ScienceNature Communications
- 2018

It is often advantageous to transform a strongly nonlinear system into a linear one in order to simplify its analysis for prediction and control, so the authors combine dynamical systems with deep learning to identify these hard-to-find transformations.

### Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems

- Computer ScienceArXiv
- 2018

This work proposes a deep learning-based strategy for nonlinear model reduction that is inspired by projection-based model reduction where the idea is to identify some optimal low-dimensional representation and evolve it in time.

### Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders

- Computer ScienceJ. Comput. Phys.
- 2020

### Deep learning of dynamics and signal-noise decomposition with time-stepping constraints

- Computer ScienceJ. Comput. Phys.
- 2019

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

- PhysicsArXiv
- 2019

### LQResNet: A Deep Neural Network Architecture for Learning Dynamic Processes

- Computer ScienceArXiv
- 2021

This work suggests combining the operator inference with certain deep neural network approaches to infer the unknown nonlinear dynamics of the system and demonstrates that the proposed methodology accomplishes the desired tasks for dynamics processes encountered in neural dynamics and the glycolytic oscillator.

### Operator Inference and Physics-Informed Learning of Low-Dimensional Models for Incompressible Flows

- Computer ScienceETNA - Electronic Transactions on Numerical Analysis
- 2021

This work utilizes the intrinsic structure of the Navier-Stokes equations for incompressible flows and shows that learning dynamics of the velocity and pressure can be decoupled, thus leading to an efficient operator inference approach for learning the underlying dynamics of incompressable flows.

### Discovering governing equations from data by sparse identification of nonlinear dynamical systems

- Computer ScienceProceedings of the National Academy of Sciences
- 2016

This work develops a novel framework to discover governing equations underlying a dynamical system simply from data measurements, leveraging advances in sparsity techniques and machine learning and using sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data.

### Dynamic mode decomposition - data-driven modeling of complex systems

- Computer Science
- 2016

This first book to address the DMD algorithm presents a pedagogical and comprehensive approach to all aspects of DMD currently developed or under development, and blends theoretical development, example codes, and applications to showcase the theory and its many innovations and uses.

### Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms

- Computer ScienceArXiv
- 2020