Learning solution operator of dynamical systems with diffusion maps kernel ridge regression
This work addresses the challenge of reliable and efficient learning for complex dynamical systems, offering a geometry-aware approach that improves predictive performance.
The authors tackled the problem of long-term prediction for complex dynamical systems by proposing a kernel ridge regression framework with a diffusion maps kernel, which consistently outperformed state-of-the-art methods in accuracy and data efficiency across various systems.
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.