# Diffeomorphically Learning Stable Koopman Operators

arXiv: https://arxiv.org/abs/2112.04085 · HTML: https://mlsanity.com/paper/2112.04085/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 24 references that are in the map.
- First posted on arXiv 8 Dec 2021.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability](https://mlsanity.com/paper/1906.03663/) (2019)
- [Koopman Operator Dynamical Models: Learning, Analysis and Control](https://mlsanity.com/paper/2102.02522/) (2021)
- [Optimal construction of Koopman eigenfunctions for prediction and control](https://mlsanity.com/paper/1810.08733/) (2018)
- [Learning Stable Koopman Embeddings](https://mlsanity.com/paper/2110.06509/) (2021)
- [Learning the Koopman Eigendecomposition: A Diffeomorphic Approach](https://mlsanity.com/paper/2110.07786/) (2021)
- [On the equivalence of contraction and Koopman approaches for nonlinear stability and control](https://mlsanity.com/paper/2103.15033/) (2021)

## Built on by (most cited first, 13 of 13)

- [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023)
- [Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees](https://mlsanity.com/paper/2402.08090/) (2024)
- [Neural Koopman prior for data assimilation](https://mlsanity.com/paper/2309.05317/) (2023)
- [Augmented Invertible Koopman Autoencoder for long-term time series forecasting](https://mlsanity.com/paper/2503.12930/) (2025)
- [Fusion Dynamical Systems with Machine Learning in Imitation Learning: A Comprehensive Overview](https://mlsanity.com/paper/2403.19916/) (2024)
- [Temporal Forward-Backward Consistency, Not Residual Error, Measures the Prediction Accuracy of Extended Dynamic Mode Decomposition](https://mlsanity.com/paper/2207.07719/) (2022)
- [Data-Driven Encoding: A New Numerical Method for Computation of the Koopman Operator](https://mlsanity.com/paper/2301.06542/) (2023)
- [Learning Noise-Robust Stable Koopman Operator for Control with Hankel DMD](https://mlsanity.com/paper/2408.06607/) (2024)
- [KoopMotion: Learning Almost Divergence Free Koopman Flow Fields for Motion Planning](https://mlsanity.com/paper/2509.09074/) (2025)
- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [Learned Lifted Linearization Applied to Unstable Dynamic Systems Enabled by Koopman Direct Encoding](https://mlsanity.com/paper/2210.13602/) (2022)
- [Koopman-based Deep Learning for Nonlinear System Estimation](https://mlsanity.com/paper/2405.00627/) (2024)
- [Learning Geometrically-Informed Lyapunov Functions with Deep Diffeomorphic RBF Networks](https://mlsanity.com/paper/2504.02607/) (2025)

## Abstract (arXiv)

System representations inspired by the infinite-dimensional Koopman operator (generator) are increasingly considered for predictive modeling. Due to the operator's linearity, a range of nonlinear systems admit linear predictor representations - allowing for simplified prediction, analysis and control. However, finding meaningful finite-dimensional representations for prediction is difficult as it involves determining features that are both Koopman-invariant (evolve linearly under the dynamics) as well as relevant (spanning the original state) - a generally unsupervised problem. In this work, we present Koopmanizing Flows - a novel continuous-time framework for supervised learning of linear predictors for a class of nonlinear dynamics. In our model construction a latent diffeomorphically related linear system unfolds into a linear predictor through the composition with a monomial basis. The lifting, its linear dynamics and state reconstruction are learned simultaneously, while an unconstrained parameterization of Hurwitz matrices ensures asymptotic stability regardless of the operator approximation accuracy. The superior efficacy of Koopmanizing Flows is demonstrated in comparison to a state-of-the-art method on the well-known LASA handwriting benchmark.