# Deep Learning of Chaotic Systems from Partially-Observed Data

arXiv: https://arxiv.org/abs/2205.08384 · HTML: https://mlsanity.com/paper/2205.08384/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 65 references that are in the map.
- First posted on arXiv 12 May 2022.

## Builds on (most cited first, 25 of 25)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data](https://mlsanity.com/paper/1908.03190/) (2019)
- [Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space](https://mlsanity.com/paper/2106.03603/) (2021)
- [Data-driven Discovery of Closure Models](https://mlsanity.com/paper/1803.09318/) (2018)
- [Learning reduced systems via deep neural networks with memory](https://mlsanity.com/paper/2003.09451/) (2020)
- [Synthesis of recurrent neural networks for dynamical system simulation](https://mlsanity.com/paper/1512.05702/) (2015)
- [Data-driven learning of non-autonomous systems](https://mlsanity.com/paper/2006.02392/) (2020)
- [Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learning](https://mlsanity.com/paper/2203.03458/) (2022)
- [Recurrent Neural Networks for Partially Observed Dynamical Systems](https://mlsanity.com/paper/2109.11629/) (2021)
- [Data-driven modelling of nonlinear dynamics by barycentric coordinates and memory](https://mlsanity.com/paper/2112.06742/) (2021)

## Built on by (most cited first, 9 of 9)

- [Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarks](https://mlsanity.com/paper/2307.11013/) (2023)
- [Deep-OSG: Deep Learning of Operators in Semigroup](https://mlsanity.com/paper/2302.03358/) (2023)
- [DUE: A Deep Learning Framework and Library for Modeling Unknown Equations](https://mlsanity.com/paper/2504.10373/) (2025)
- [Targeted Digital Twin via Flow Map Learning and Its Application to Fluid Dynamics](https://mlsanity.com/paper/2510.07549/) (2025)
- [Learning Chaotic Systems and Long-Term Predictions with Neural Jump ODEs](https://mlsanity.com/paper/2407.18808/) (2024)
- [Data-driven synchronization-avoiding algorithms in the explicit distributed structural analysis of soft tissue](https://mlsanity.com/paper/2207.02194/) (2022)
- [Learning Fine Scale Dynamics from Coarse Observations via Inner Recurrence](https://mlsanity.com/paper/2206.01807/) (2022)
- [Data-Augmented Few-Shot Neural Emulator for Computer-Model System Identification](https://mlsanity.com/paper/2508.19441/) (2025)
- [More Experts, Worse Dynamics: Inverse Scaling and Spectral Bias in Mixture-of-Experts State-Space Models](https://mlsanity.com/paper/2608.21840/) (2026)

## Abstract (arXiv)

Recently, a general data driven numerical framework has been developed for learning and modeling of unknown dynamical systems using fully- or partially-observed data. The method utilizes deep neural networks (DNNs) to construct a model for the flow map of the unknown system. Once an accurate DNN approximation of the flow map is constructed, it can be recursively executed to serve as an effective predictive model of the unknown system. In this paper, we apply this framework to chaotic systems, in particular the well-known Lorenz 63 and 96 systems, and critically examine the predictive performance of the approach. A distinct feature of chaotic systems is that even the smallest perturbations will lead to large (albeit bounded) deviations in the solution trajectories. This makes long-term predictions of the method, or any data driven methods, questionable, as the local model accuracy will eventually degrade and lead to large pointwise errors. Here we employ several other qualitative and quantitative measures to determine whether the chaotic dynamics have been learned. These include phase plots, histograms, autocorrelation, correlation dimension, approximate entropy, and Lyapunov exponent. Using these measures, we demonstrate that the flow map based DNN learning method is capable of accurately modeling chaotic systems, even when only a subset of the state variables are available to the DNNs. For example, for the Lorenz 96 system with 40 state variables, when data of only 3 variables are available, the method is able to learn an effective DNN model for the 3 variables and produce accurately the chaotic behavior of the system.