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

2025/12/19 by Jiwoo Song, Daning Huang, Song, Jiwoo +3
Physics and Astronomy · Computer Science · #Model Reduction and Neural Networks #Neural Networks and Reservoir Computing #Micro and Nano Robotics

paper · doi:10.48550/arxiv.2512.17203

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.

Citations

• Physics-Infused Reduced-Order Modeling for Analysis of Multi-Layered Hypersonic Thermal Protection Systems
• Learning dynamical systems with hit-and-run random feature maps
• Modal Analysis of Spatiotemporal Data via Multivariate Gaussian Process Regression
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• Lyapunov exponents of the Kuramoto-Sivashinsky PDE
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• Deterministic Nonperiodic Flow

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