# Stability analysis of chaotic systems in latent spaces

arXiv: https://arxiv.org/abs/2410.00480 · HTML: https://mlsanity.com/paper/2410.00480/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 77 references that are in the map.
- First posted on arXiv 1 Oct 2024.

## Builds on (most cited first, 17 of 17)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [Deep learning to discover and predict dynamics on an inertial manifold](https://mlsanity.com/paper/2001.04263/) (2019)
- [Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.00060/) (2021)
- [Control of chaotic systems by Deep Reinforcement Learning](https://mlsanity.com/paper/1906.07672/) (2019)
- [Robust Optimization and Validation of Echo State Networks for learning chaotic dynamics](https://mlsanity.com/paper/2103.03174/) (2021)
- [Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoders](https://mlsanity.com/paper/2307.15288/) (2023)
- [Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder](https://mlsanity.com/paper/2310.10745/) (2023)
- [Predicting the temporal dynamics of turbulent channels through deep learning](https://mlsanity.com/paper/2203.00974/) (2022)
- [Reconstruction, forecasting, and stability of chaotic dynamics from partial data](https://mlsanity.com/paper/2305.15111/) (2023)
- [On interpretability and proper latent decomposition of autoencoders](https://mlsanity.com/paper/2211.08345/) (2022)
- [Forecasting through deep learning and modal decomposition in two-phase concentric jets](https://mlsanity.com/paper/2212.12731/) (2022)
- [Decoder Decomposition for the Analysis of the Latent Space of Nonlinear Autoencoders With Wind-Tunnel Experimental Data](https://mlsanity.com/paper/2404.19660/) (2024)

## Built on by (most cited first, 7 of 7)

- [Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2501.13329/) (2025)
- [Robust quantum reservoir computers for forecasting chaotic dynamics: generalized synchronization and stability](https://mlsanity.com/paper/2506.22335/) (2025)
- [Online model learning with data-assimilated reservoir computers](https://mlsanity.com/paper/2504.16767/) (2025)
- [UP-dROM : Uncertainty-Aware and Parametrised dynamic Reduced-Order Model, application to unsteady flows](https://mlsanity.com/paper/2503.23236/) (2025)
- [HFNO: an interpretable data-driven decomposition strategy for turbulent flows](https://mlsanity.com/paper/2511.01535/) (2025)
- [Convolutional Symmetric AutoEncoders: enhancing latent stability via differential geometry](https://mlsanity.com/paper/2607.00669/) (2026)
- [The balance between compactness and forecast accuracy of data-driven latent-space reduced-order models in controlled wake flows](https://mlsanity.com/paper/2607.24569/) (2026)

## Abstract (arXiv)

Partial differential equations, and their chaotic solutions, are pervasive in the modelling of complex systems in engineering, science, and beyond. Data-driven methods can find solutions to partial differential equations with a divide-and-conquer strategy: The solution is sought in a latent space, on which the temporal dynamics are inferred (``latent-space'' approach). This is achieved by, first, compressing the data with an autoencoder, and, second, inferring the temporal dynamics with recurrent neural networks. The overarching goal of this paper is to show that a latent-space approach can not only infer the solution of a chaotic partial differential equation, but it can also predict the stability properties of the physical system. First, we employ the convolutional autoencoder echo state network (CAE-ESN) on the chaotic Kuramoto-Sivashinsky equation for various chaotic regimes. We show that the CAE-ESN (i) finds a low-dimensional latent-space representation of the observations and (ii) accurately infers the Lyapunov exponents and covariant Lyapunov vectors (CLVs) in this low-dimensional manifold for different attractors. Second, we extend the CAE-ESN to a turbulent flow, comparing the Lyapunov spectrum to estimates obtained from Jacobian-free methods. A latent-space approach based on the CAE-ESN effectively produces a latent space that preserves the key properties of the chaotic system, such as Lyapunov exponents and CLVs, thus retaining the geometric structure of the attractor. The latent-space approach based on the CAE-ESN is a reduced-order model that accurately predicts the dynamics of the chaotic system, or, alternatively, it can be used to infer stability properties of chaotic systems from data.