A density-based manifold learning framework for reconstructing high-dimensional dynamical systems with outliers

Data-driven dynamic dimensionality reduction is crucial for uncovering the low-dimensional dynamics of complex systems. However, existing methods often lack robustness and adaptability in parameter selection. To address these limitations, this study proposes a novel framework: Density-based Decomposition on Manifold with Autoencoder Reduction (DDMAR). It enhances robustness to outliers and parameter sensitivity, reduces the need for manual parameter tuning, and maintains efficient reconstruction with smooth global continuity. DDMAR first employs density-based spatial clustering (DBSCAN) to decompose the manifold into multiple overlapping atlases, effectively capturing the dynamic evolution of the system. A semi-automated clustering parameter selection strategy, integrating the Calinski–Harabasz index with outlier analysis, optimises the decomposition process and reduces reliance on exhaustive manual parameter adjustment. Dimensionality reduction and reconstruction are then achieved using an autoencoder, which is further integrated with a feedforward neural network to model the low-dimensional dynamics. The effectiveness of DDMAR is validated on three benchmark models: the Torus equation, Euler Attitude equation, and Kuramoto–Sivashinsky equation. Compared to traditional manifold-based approaches, DDMAR's parameter selection strategy and core point principle greatly improve the reliability and stability of manifold learning for dynamic systems. The method also demonstrates superior robustness to outliers, enabling accurate extraction of intrinsic coordinates and minimising reconstruction error. Notably, the atlas-based decomposition strategy ensures smoother transitions in reconstructed dynamics. By integrating manifold learning with autoencoder-based dimensionality reduction, this study establishes a robust and adaptable framework for dimensionality reduction and reconstruction of complex, high-dimensional dynamical systems with outliers.