Learning solution operator of dynamical systems with diffusion maps kernel ridge regression

TL;DR

This paper introduces a diffusion maps kernel ridge regression method that leverages intrinsic geometric structures of dynamical systems for improved long-term prediction, outperforming existing methods in accuracy and data efficiency.

Contribution

The paper presents a novel DM-KRR approach that uses diffusion maps to incorporate geometric information into kernel ridge regression for dynamical systems prediction.

Findings

DM-KRR outperforms state-of-the-art methods in accuracy.

DM-KRR is more data-efficient across various systems.

Geometry-aware modeling improves long-term predictions.

Abstract

In this work, we propose a simple kernel ridge regression (KRR) framework with a dynamic-aware validation strategy for long-term prediction of complex dynamical systems. By employing a data-driven kernel derived from diffusion maps, the proposed Diffusion Maps Kernel Ridge Regression (DM-KRR) method implicitly adapts to the intrinsic geometry of the system's invariant set, without requiring explicit manifold reconstruction or attractor modeling, procedures that often limit predictive performance. Across a broad range of systems, including smooth manifolds, chaotic attractors, and high-dimensional spatiotemporal flows, DM-KRR consistently outperforms state-of-the-art random feature, neural-network and operator-learning methods in both accuracy and data efficiency. These findings underscore that long-term predictive skill depends not only on model expressiveness, but critically on respecting the geometric constraints encoded in the data through dynamically consistent model selection. Together, simplicity, geometry awareness, and strong empirical performance point to a promising path for reliable and efficient learning of complex dynamical systems.