# Linearly-Recurrent Autoencoder Networks for Learning Dynamics

arXiv: https://arxiv.org/abs/1712.01378 · HTML: https://mlsanity.com/paper/1712.01378/

## Key facts

- Cited by 145 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2017.

## Built on by (most cited first, 30 of 145)

- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [Generalized Teacher Forcing for Learning Chaotic Dynamics](https://mlsanity.com/paper/2306.04406/) (2023)
- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Deeptime: a Python library for machine learning dynamical models from time series data](https://mlsanity.com/paper/2110.15013/) (2021)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [Deep networks for system identification: a Survey](https://mlsanity.com/paper/2301.12832/) (2023)
- [Out-of-Domain Generalization in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2402.18377/) (2024)
- [Koopman neural operator as a mesh-free solver of non-linear partial differential equations](https://mlsanity.com/paper/2301.10022/) (2023)
- [Learning continuous models for continuous physics](https://mlsanity.com/paper/2202.08494/) (2022)
- [A Comprehensive Review of Digital Twin -- Part 1: Modeling and Twinning Enabling Technologies](https://mlsanity.com/paper/2208.14197/) (2022)
- [Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds](https://mlsanity.com/paper/2201.04976/) (2022)
- [Multifactor Sequential Disentanglement via Structured Koopman Autoencoders](https://mlsanity.com/paper/2303.17264/) (2023)
- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2303.17078/) (2023)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2501.13329/) (2025)
- [True Zero-Shot Inference of Dynamical Systems Preserving Long-Term Statistics](https://mlsanity.com/paper/2505.13192/) (2025)
- [Almost-Linear RNNs Yield Highly Interpretable Symbolic Codes in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2410.14240/) (2024)
- [Learning Stable Koopman Embeddings](https://mlsanity.com/paper/2110.06509/) (2021)
- [Multi-fidelity reduced-order surrogate modeling](https://mlsanity.com/paper/2309.00325/) (2023)
- [Learning invariant representations of time-homogeneous stochastic dynamical systems](https://mlsanity.com/paper/2307.09912/) (2023)
- [Temporally Consistent Koopman Autoencoders for Forecasting Dynamical Systems](https://mlsanity.com/paper/2403.12335/) (2024)
- [Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution](https://mlsanity.com/paper/2103.14629/) (2021)
- [On latent dynamics learning in nonlinear reduced order modeling](https://mlsanity.com/paper/2408.15183/) (2024)
- [Applying Machine Learning to Study Fluid Mechanics](https://mlsanity.com/paper/2110.02083/) (2021)
- [Koopman operator learning using invertible neural networks](https://mlsanity.com/paper/2306.17396/) (2023)
- [Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder](https://mlsanity.com/paper/2310.10745/) (2023)

## Abstract (arXiv)

This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural-network approximations of the underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD) provides a useful data-driven approximation of the Koopman operator for analyzing dynamical systems. This paper addresses a fundamental problem associated with EDMD: a trade-off between representational capacity of the dictionary and over-fitting due to insufficient data. A new neural network architecture combining an autoencoder with linear recurrent dynamics in the encoded state is used to learn a low-dimensional and highly informative Koopman-invariant subspace of observables. A method is also presented for balanced model reduction of over-specified EDMD systems in feature space. Nonlinear reconstruction using partially linear multi-kernel regression aims to improve reconstruction accuracy from the low-dimensional state when the data has complex but intrinsically low-dimensional structure. The techniques demonstrate the ability to identify Koopman eigenfunctions of the unforced Duffing equation, create accurate low-dimensional models of an unstable cylinder wake flow, and make short-time predictions of the chaotic Kuramoto-Sivashinsky equation.