# Deep Learning for Stability Analysis of a Freely Vibrating Sphere at Moderate Reynolds Number

arXiv: https://arxiv.org/abs/2112.09858 · HTML: https://mlsanity.com/paper/2112.09858/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 60 references that are in the map.
- First posted on arXiv 18 Dec 2021.

## Builds on (most cited first, 16 of 16)

- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [A Critical Review of Recurrent Neural Networks for Sequence Learning](https://mlsanity.com/paper/1506.00019/) (2015)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [Koopman operator-based model reduction for switched-system control of PDEs](https://mlsanity.com/paper/1710.06759/) (2017)
- [Assessment of unsteady flow predictions using hybrid deep learning based reduced order models](https://mlsanity.com/paper/2009.04396/) (2020)
- [Deep learning algorithm for data-driven simulation of noisy dynamical system](https://mlsanity.com/paper/1802.08323/) (2018)
- [Linear System Identification in a Nonlinear Setting - Nonparametric analysis of the nonlinear distortions and their impact on the best linear approximation](https://mlsanity.com/paper/1804.09587/) (2018)
- [Decomposition of Nonlinear Dynamical Systems Using Koopman Gramians](https://mlsanity.com/paper/1710.01719/) (2017)

## Abstract (arXiv)

In this paper, we present a deep learning-based reduced-order model (DL-ROM) for the stability prediction of unsteady 3D fluid-structure interaction systems. The proposed DL-ROM has the format of a nonlinear state-space model and employs a recurrent neural network with long short-term memory (LSTM). We consider a canonical fluid-structure system of an elastically-mounted sphere coupled with incompressible fluid flow in a state-space format. We develop a nonlinear data-driven coupling for predicting unsteady forces and vortex-induced vibration (VIV) lock-in of the freely vibrating sphere in a transverse direction. We design an input-output relationship as a temporal sequence of force and displacement datasets for a low-dimensional approximation of the fluid-structure system. Based on the prior knowledge of the VIV lock-in process, the input function contains a range of frequencies and amplitudes, which enables an efficient DL-ROM without the need for a massive training dataset for the low-dimensional modeling. Once trained, the network provides a nonlinear mapping of input-output dynamics that can predict the coupled fluid-structure dynamics for a longer horizon via the feedback process. By integrating the LSTM network with the eigensystem realization algorithm (ERA), we construct a data-driven state-space model for the reduced-order stability analysis. We investigate the underlying mechanism and stability characteristics of VIV via an eigenvalue selection process. To understand the frequency lock-in mechanism, we study the eigenvalue trajectories for a range of the reduced oscillation frequencies and the mass ratios. Consistent with the full-order simulations, the frequency lock-in branches are accurately captured by the combined LSTM-ERA procedure. The proposed DL-ROM aligns with the development of physics-based digital twin of engineering systems involving fluid-structure interactions.