# Ensemble forecasts in reproducing kernel Hilbert space family

arXiv: https://arxiv.org/abs/2207.14653 · HTML: https://mlsanity.com/paper/2207.14653/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 99 references that are in the map.
- First posted on arXiv 29 Jul 2022.

## Builds on (most cited first, 22 of 22)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Koopman Operator Dynamical Models: Learning, Analysis and Control](https://mlsanity.com/paper/2102.02522/) (2021)
- [Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions](https://mlsanity.com/paper/1604.05251/) (2016)
- [Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization](https://mlsanity.com/paper/2001.06270/) (2020)
- [Learning Variational Data Assimilation Models and Solvers](https://mlsanity.com/paper/2007.12941/) (2020)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation](https://mlsanity.com/paper/2007.07383/) (2020)
- [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)
- [Learning Latent Dynamics for Partially-Observed Chaotic Systems](https://mlsanity.com/paper/1907.02452/) (2019)
- [Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods](https://mlsanity.com/paper/2103.10935/) (2021)
- [Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series](https://mlsanity.com/paper/2111.13037/) (2021)
- [Kernel Analog Forecasting: Multiscale Test Problems](https://mlsanity.com/paper/2005.06623/) (2020)
- [Sparsity-promoting algorithms for the discovery of informative Koopman invariant subspaces](https://mlsanity.com/paper/2002.10637/) (2020)
- [Kernel-based Prediction of Non-Markovian Time Series](https://mlsanity.com/paper/2007.04286/) (2020)
- [Eigenvalues of Autocovariance Matrix: A Practical Method to Identify the Koopman Eigenfrequencies](https://mlsanity.com/paper/2107.01948/) (2021)
- [Bridging Koopman Operator and time-series auto-correlation based Hilbert-Schmidt operator](https://mlsanity.com/paper/2202.08755/) (2022)

## Built on by (most cited first, 1 of 1)

- [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025)

## Abstract (arXiv)

A methodological framework for ensemble-based estimation and simulation of high dimensional dynamical systems such as the oceanic or atmospheric flows is proposed. To that end, the dynamical system is embedded in a family of reproducing kernel Hilbert spaces (RKHS) with kernel functions driven by the dynamics. In the RKHS family, the Koopman and Perron-Frobenius operators are unitary and uniformly continuous. This property warrants they can be expressed in exponential series of diagonalizable bounded evolution operators defined from their infinitesimal generators. Access to Lyapunov exponents and to exact ensemble based expressions of the tangent linear dynamics are directly available as well. The RKHS family enables us the devise of strikingly simple ensemble data assimilation methods for trajectory reconstructions in terms of constant-in-time linear combinations of trajectory samples. Such an embarrassingly simple strategy is made possible through a fully justified superposition principle ensuing from several fundamental theorems.