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

2025/12/19 by Jiwoo Song, Daning Huang, Song, Jiwoo +3
Computer Science · Physics and Astronomy · #Attractor #Chaotic #Diffusion map #Dynamical systems theory #Invariant (physics) #Kernel (algebra) #Kernel method #Micro and Nano Robotics #Model Reduction and Neural Networks #Neural Networks and Reservoir Computing #Operator (biology) #Range (aeronautics) #Ridge

paper · open access · doi:10.48550/arxiv.2512.17203

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/12/19 · openalex created_date 2025/12/23 · openalex updated_date 2026/07/28

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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• A Framework for Machine Learning of Model Error in Dynamical Systems
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• Sparse Cholesky factorization by Kullback-Leibler minimization
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• Learning Discrepancy Models From Experimental Data
• Lyapunov exponents of the Kuramoto-Sivashinsky PDE
• Variable Bandwidth Diffusion Kernels
• Deterministic Nonperiodic Flow

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