# Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem

arXiv: https://arxiv.org/abs/2012.00338 · HTML: https://mlsanity.com/paper/2012.00338/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Dec 2020.

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

- [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)
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- [Bridging Algorithmic Information Theory and Machine Learning: A New Approach to Kernel Learning](https://mlsanity.com/paper/2311.12624/) (2023)
- [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)
- [Simplicity bias, algorithmic probability, and the random logistic map](https://mlsanity.com/paper/2401.00593/) (2023)
- [A Dynamics-Informed Gaussian Process Framework for 2D Stochastic Navier-Stokes via Quasi-Gaussianity](https://mlsanity.com/paper/2511.21281/) (2025)
- [Adaptive Kernel Selection for Kernelized Diffusion Maps](https://mlsanity.com/paper/2604.18402/) (2026)
- [Dictionary learning for Kernel EDMD](https://mlsanity.com/paper/2604.25572/) (2026)

## Abstract (arXiv)

For dynamical systems with a non hyperbolic equilibrium, it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to the equilibrium point and to obtain meaningful predictions of its behavior by analyzing a reduced order system on the so-called center manifold. Since the center manifold is usually not known, good approximation methods are important as the center manifold theorem states that the stability properties of the origin of the reduced order system are the same as those of the origin of the full order system. In this work, we establish a data-based version of the center manifold theorem that works by considering an approximation in place of an exact manifold. Also the error between the approximated and the original reduced dynamics are quantified. We then use an apposite data-based kernel method to construct a suitable approximation of the manifold close to the equilibrium, which is compatible with our general error theory. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used as a training set. The method is tested on different examples which show promising performance and good accuracy.