# Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems

arXiv: https://arxiv.org/abs/2301.10321 · HTML: https://mlsanity.com/paper/2301.10321/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 62 references that are in the map.
- First posted on arXiv 24 Jan 2023.

## Builds on (most cited first, 18 of 18)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Chaos as an interpretable benchmark for forecasting and data-driven modelling](https://mlsanity.com/paper/2110.05266/) (2021)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Computational Graph Completion](https://mlsanity.com/paper/2110.10323/) (2021)
- [Learning dynamical systems from data: a simple cross-validation perspective](https://mlsanity.com/paper/2007.05074/) (2020)
- [Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)](https://mlsanity.com/paper/2106.01510/) (2021)
- [Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods](https://mlsanity.com/paper/2103.10935/) (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)
- [A Note on Kernel Methods for Multiscale Systems with Critical Transitions](https://mlsanity.com/paper/1804.09415/) (2018)
- [Ensemble Kalman methods for high-dimensional hierarchical dynamic space-time models](https://mlsanity.com/paper/1704.06988/) (2017)
- [Deep regularization and direct training of the inner layers of Neural Networks with Kernel Flows](https://mlsanity.com/paper/2002.08335/) (2020)
- [Data-driven prediction of a multi-scale Lorenz 96 chaotic system using deep learning methods: Reservoir computing, ANN, and RNN-LSTM](https://mlsanity.com/paper/1906.08829/) (2019)
- [Learning "best" kernels from data in Gaussian process regression. With application to aerodynamics](https://mlsanity.com/paper/2206.02563/) (2022)
- [Sparse dynamical system identification with simultaneous structural parameters and initial condition estimation](https://mlsanity.com/paper/2204.10472/) (2022)

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

- [Out-of-Domain Generalization in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2402.18377/) (2024)
- [Zero-shot forecasting of chaotic systems](https://mlsanity.com/paper/2409.15771/) (2024)
- [Context parroting: A simple but tough-to-beat baseline for foundation models in scientific machine learning](https://mlsanity.com/paper/2505.11349/) (2025)
- [A scalable generative model for dynamical system reconstruction from neuroimaging data](https://mlsanity.com/paper/2411.02949/) (2024)
- [Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis](https://mlsanity.com/paper/2503.01036/) (2025)
- [Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator](https://mlsanity.com/paper/2412.16588/) (2024)
- [Kernel Sum of Squares for Data Adapted Kernel Learning of Dynamical Systems from Data: A global optimization approach](https://mlsanity.com/paper/2408.06465/) (2024)
- [Bridging Algorithmic Information Theory and Machine Learning: A New Approach to Kernel Learning](https://mlsanity.com/paper/2311.12624/) (2023)
- [Decoding Mean Field Games from Population and Environment Observations By Gaussian Processes](https://mlsanity.com/paper/2312.06625/) (2023)
- [Learning Surrogate Potential Mean Field Games via Gaussian Processes: A Data-Driven Approach to Ill-Posed Inverse Problems](https://mlsanity.com/paper/2502.11506/) (2025)
- [Recurrent Deep Kernel Learning of Dynamical Systems](https://mlsanity.com/paper/2405.19785/) (2024)
- [Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation](https://mlsanity.com/paper/2603.01077/) (2026)
- [Kernel Methods for Some Transport Equations with Application to Learning Kernels for the Approximation of Koopman Eigenfunctions: A Unified Approach via Variational Methods, Green's Functions and the Method of Characteristics](https://mlsanity.com/paper/2603.06872/) (2026)
- [Gaussian process policy iteration with additive Schwarz acceleration for forward and inverse HJB and mean field game problems](https://mlsanity.com/paper/2505.00909/) (2025)
- [Adaptive Kernel Selection for Kernelized Diffusion Maps](https://mlsanity.com/paper/2604.18402/) (2026)

## Abstract (arXiv)

Regressing the vector field of a dynamical system from a finite number of observed states is a natural way to learn surrogate models for such systems. A simple and interpretable way to learn a dynamical system from data is to interpolate its vector-field with a data-adapted kernel which can be learned by using Kernel Flows. The method of Kernel Flows is a trainable machine learning method that learns the optimal parameters of a kernel based on the premise that a kernel is good if there is no significant loss in accuracy if half of the data is used. The objective function could be a short-term prediction or some other objective for other variants of Kernel Flows). However, this method is limited by the choice of the base kernel. In this paper, we introduce the method of Sparse Kernel Flows  in order to learn the ``best'' kernel by starting from a large dictionary of kernels. It is based on sparsifying a kernel that is a linear combination of elemental kernels. We apply this approach to a library of 132 chaotic systems.