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

TL;DR

The paper introduces DM-KRR, a simple kernel ridge regression approach that uses a diffusion maps-based kernel to learn the solution operator of dynamical systems. By coupling this geometry-aware kernel with a dynamics-aware validation strategy, the method achieves superior long-term predictive accuracy and data efficiency across manifolds, chaotic attractors, and high-dimensional flows. The key contribution is showing that respecting the intrinsic geometry of forward invariant sets—without explicit manifold reconstruction—substantially improves long-horizon predictions, outperforming neural, random-feature, and operator-learning baselines. The work highlights the importance of model selection criteria and geometric alignment in data-driven dynamics, with practical implications for scalable, reliable forecasting in complex systems.

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.

Figures (17)

• Figure 1: Convergence study on the torus dynamics. The RMSE's are evaluated on the 32000 test trajectories. The markers and shades show the mean and min-max bounds of RMSE's, respectively. Consistently over all the ambient dimensions, DM exhibits a faster convergence rate than the RBF.
• Figure 2: Representative predictions of DM and RBF models for Lorenz and K-S chaotic systems. Intrinsic coordinates are constructed by the first 3 eigenfunctions of the DM kernel function. The DM clearly outperforms RBF in terms of VPT in both cases.
• Figure 3: The prediction results for the Kuramoto-Sivashinsky travelling dynamics, comparing the DM and RBF models against four baseline methods. The DM and RBF models perform drastically better than the baselines over the long timescale, while DM is one order of magnitude better than RBF.
• Figure 4: The prediction results for the pitching and plunging plate at $t=80$, comparing the DM, RBF, and ResDMD models. The RBF model fails likely due to the data sparsity, while DM outperforms ResDMD in accuracy and maintains the same level of accuracy over a long prediction horizon.
• Figure 5: Visualization of the transformation from a unit sphere to a torus.