# Temporally Consistent Koopman Autoencoders for Forecasting Dynamical Systems

arXiv: https://arxiv.org/abs/2403.12335 · HTML: https://mlsanity.com/paper/2403.12335/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 19 Mar 2024.

## Builds on (most cited first, 21 of 21)

- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Regularization With Stochastic Transformations and Perturbations for Deep Semi-Supervised Learning](https://mlsanity.com/paper/1606.04586/) (2016)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [AntisymmetricRNN: A Dynamical System View on Recurrent Neural Networks](https://mlsanity.com/paper/1902.09689/) (2019)
- [Stable Recurrent Models](https://mlsanity.com/paper/1805.10369/) (2018)
- [Physics Guided RNNs for Modeling Dynamical Systems: A Case Study in Simulating Lake Temperature Profiles](https://mlsanity.com/paper/1810.13075/) (2018)
- [From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction](https://mlsanity.com/paper/2004.00574/) (2020)
- [Revisiting Consistency Regularization for Semi-Supervised Learning](https://mlsanity.com/paper/2112.05825/) (2021)
- [Optimizing Neural Networks via Koopman Operator Theory](https://mlsanity.com/paper/2006.02361/) (2020)
- [Non-normal Recurrent Neural Network (nnRNN): learning long time dependencies while improving expressivity with transient dynamics](https://mlsanity.com/paper/1905.12080/) (2019)
- [Consistency Regularization Can Improve Robustness to Label Noise](https://mlsanity.com/paper/2110.01242/) (2021)
- [Data-Driven Model Predictive Control using Interpolated Koopman Generators](https://mlsanity.com/paper/2003.07094/) (2020)
- [Equivariant Deep Dynamical Model for Motion Prediction](https://mlsanity.com/paper/2111.01892/) (2021)
- [SE(3) Koopman-MPC: Data-driven Learning and Control of Quadrotor UAVs](https://mlsanity.com/paper/2305.03868/) (2023)

## Built on by (most cited first, 15 of 15)

- [Learning Robust Spectral Dynamics for Temporal Domain Generalization](https://mlsanity.com/paper/2505.12585/) (2025)
- [When Graph Neural Networks Meet Dynamic Mode Decomposition](https://mlsanity.com/paper/2410.05593/) (2024)
- [Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoders](https://mlsanity.com/paper/2505.13569/) (2025)
- [Beyond Ensembles: Simulating All-Atom Protein Dynamics in a Learned Latent Space](https://mlsanity.com/paper/2509.02196/) (2025)
- [Latent Autoencoder Ensemble Kalman Filter for Nonlinear Data assimilation](https://mlsanity.com/paper/2603.06752/) (2026)
- [Spectrally Informed Learning of Fluid Flows](https://mlsanity.com/paper/2408.14407/) (2024)
- [KoopGen: Koopman Generator Networks for Representing and Predicting Dynamical Systems with Continuous Spectra](https://mlsanity.com/paper/2602.14011/) (2026)
- [Reduced-Order Surrogates for Forced Flexible Mesh Coastal-Ocean Models](https://mlsanity.com/paper/2602.05416/) (2026)
- [Surrogate models for Rock-Fluid Interaction: A Grid-Size-Invariant Approach](https://mlsanity.com/paper/2602.22188/) (2026)
- [On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations](https://mlsanity.com/paper/2604.03086/) (2026)
- [Parametric Interpolation of Dynamic Mode Decomposition for Predicting Nonlinear Systems](https://mlsanity.com/paper/2604.12103/) (2026)
- [Learning the Koopman Operator using Attention Free Transformers](https://mlsanity.com/paper/2606.23957/) (2026)
- [RAMP: Recognition parametrisation by Amortised Message Passing](https://mlsanity.com/paper/2607.18883/) (2026)
- [Learning Prognostic Variables for AI Convective Parameterizations via Symbolic Distillation](https://mlsanity.com/paper/2609.24882/) (2026)
- [Variational Augmented Invertible Koopman Autoencoder for probabilistic time series forecasting](https://mlsanity.com/paper/2609.37435/) (2026)

## Abstract (arXiv)

Absence of sufficiently high-quality data often poses a key challenge in data-driven modeling of high-dimensional spatio-temporal dynamical systems. Koopman Autoencoders (KAEs) harness the expressivity of deep neural networks (DNNs), the dimension reduction capabilities of autoencoders, and the spectral properties of the Koopman operator to learn a reduced-order feature space with simpler, linear dynamics. However, the effectiveness of KAEs is hindered by limited and noisy training datasets, leading to poor generalizability. To address this, we introduce the temporally consistent Koopman autoencoder (tcKAE), designed to generate accurate long-term predictions even with limited and noisy training data. This is achieved through a consistency regularization term that enforces prediction coherence across different time steps, thus enhancing the robustness and generalizability of tcKAE over existing models. We provide analytical justification for this approach based on Koopman spectral theory and empirically demonstrate tcKAE's superior performance over state-of-the-art KAE models across a variety of test cases, including simple pendulum oscillations, kinetic plasma, and fluid flow data.