# Dictionary learning for Kernel EDMD

arXiv: https://arxiv.org/abs/2604.25572 · HTML: https://mlsanity.com/paper/2604.25572/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 67 references that are in the map.
- First posted on arXiv 28 Apr 2026.

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

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [Tensor Programs I: Wide Feedforward or Recurrent Neural Networks of Any Architecture are Gaussian Processes](https://mlsanity.com/paper/1910.12478/) (2019)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Finite-data error bounds for Koopman-based prediction and control](https://mlsanity.com/paper/2108.07102/) (2021)
- [Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem](https://mlsanity.com/paper/2012.00338/) (2020)
- [Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds](https://mlsanity.com/paper/1904.08622/) (2019)
- [Koopman-informed recurrent neural networks](https://mlsanity.com/paper/2410.23467/) (2024)
- [Two methods to approximate the Koopman operator with a reservoir computer](https://mlsanity.com/paper/2008.10263/) (2020)
- [Convergent Methods for Koopman Operators on Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2506.15782/) (2025)
- [Enhancing Predictive Capabilities in Data-Driven Dynamical Modeling with Automatic Differentiation: Koopman and Neural ODE Approaches](https://mlsanity.com/paper/2310.06790/) (2023)
- [Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator](https://mlsanity.com/paper/2412.16588/) (2024)
- [Online Estimation of the Koopman Operator Using Fourier Features](https://mlsanity.com/paper/2212.01503/) (2022)
- [Kernel Mode Decomposition and programmable/interpretable regression networks](https://mlsanity.com/paper/1907.08592/) (2019)
- [Bridging Algorithmic Information Theory and Machine Learning: A New Approach to Kernel Learning](https://mlsanity.com/paper/2311.12624/) (2023)
- [Kernel EDMD for data-driven nonlinear Koopman MPC with stability guarantees](https://mlsanity.com/paper/2501.08709/) (2025)

## Abstract (arXiv)

Studying nonlinear dynamical systems through their state space behavior can be challenging, and one possible alternative is to analyze them via their associated Koopman operator. This turns the nonlinear problem into a linear, infinite-dimensional one. To approximate the operator in finite dimensions, extended dynamic mode decomposition (EDMD) is a commonly used algorithm. It requires a finite list of functionals and a set of snapshots from the system to compute an approximation of the operator and its corresponding spectrum. Instead of choosing the list of functionals directly, it can be implicitly defined via kernels, a method known as kernel extended dynamic mode decomposition (kEDMD). However, one still needs to define the kernel and choose its parameter values. In this paper, we aim to streamline this process by extending dictionary learning for EDMD to kernel learning in kEDMD. By simplifying kEDMD, we show how to perform gradient-based optimization over the learnable kernel parameters and demonstrate that this method leads to useful kernels for the original kEDMD. The focus of our work is a method that takes a weighted list of kernels with randomly initialized values as input and outputs a list of kernels and parameter values suitable for approximating the Koopman operator of the underlying system. We demonstrate that unimportant kernels can be removed from the list by analyzing the weights in the weighted sum. We evaluate the method across several experiments, including the Duffing oscillator and the Kuramoto-Sivashinsky equation, showcasing the method's different strengths.