# Kernel Sum of Squares for Data Adapted Kernel Learning of Dynamical Systems from Data: A global optimization approach

arXiv: https://arxiv.org/abs/2408.06465 · HTML: https://mlsanity.com/paper/2408.06465/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 59 references that are in the map.
- First posted on arXiv 12 Aug 2024.

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

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (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)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [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)
- [Finding Global Minima via Kernel Approximations](https://mlsanity.com/paper/2012.11978/) (2020)
- [Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods](https://mlsanity.com/paper/2103.10935/) (2021)
- [One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels](https://mlsanity.com/paper/2209.12086/) (2022)
- [Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems](https://mlsanity.com/paper/2301.10321/) (2023)
- [Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series](https://mlsanity.com/paper/2111.13037/) (2021)
- [Iterative Ensemble Kalman Methods: A Unified Perspective with Some New Variants](https://mlsanity.com/paper/2010.13299/) (2020)
- [Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem](https://mlsanity.com/paper/2012.00338/) (2020)
- [A Note on Kernel Methods for Multiscale Systems with Critical Transitions](https://mlsanity.com/paper/1804.09415/) (2018)
- [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)
- [Curvature Aligned Simplex Gradient: Principled Sample Set Construction For Numerical Differentiation](https://mlsanity.com/paper/2310.12712/) (2023)

## Built on by (most cited first, 4 of 4)

- [Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator](https://mlsanity.com/paper/2412.16588/) (2024)
- [Data-efficient Kernel Methods for Learning Hamiltonian Systems](https://mlsanity.com/paper/2509.17154/) (2025)
- [Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation](https://mlsanity.com/paper/2603.01077/) (2026)
- [A Dynamics-Informed Gaussian Process Framework for 2D Stochastic Navier-Stokes via Quasi-Gaussianity](https://mlsanity.com/paper/2511.21281/) (2025)

## Abstract (arXiv)

This paper examines the application of the Kernel Sum of Squares (KSOS) method for enhancing kernel learning from data, particularly in the context of dynamical systems. Traditional kernel-based methods, despite their theoretical soundness and numerical efficiency, frequently struggle with selecting optimal base kernels and parameter tuning, especially with gradient-based methods prone to local optima. KSOS mitigates these issues by leveraging a global optimization framework with kernel-based surrogate functions, thereby achieving more reliable and precise learning of dynamical systems. Through comprehensive numerical experiments on the Logistic Map, Henon Map, and Lorentz System, KSOS is shown to consistently outperform gradient descent in minimizing the relative-ρ metric and improving kernel accuracy. These results highlight KSOS's effectiveness in predicting the behavior of chaotic dynamical systems, demonstrating its capability to adapt kernels to underlying dynamics and enhance the robustness and predictive power of kernel-based approaches, making it a valuable asset for time series analysis in various scientific fields.