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

Abstract: Many scientific and engineering systems exhibit complex nonlinear dynamics that are difficult to predict accurately over long time horizons. Although data-driven models have shown promise, their performance often deteriorates when the geometric structures governing long-term behavior are unknown or poorly represented. We demonstrate that a simple kernel ridge regression (KRR) framework, when combined with a dynamics-aware validation strategy, provides a strong baseline 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.