Low-rank physics-constrained dynamic mode decomposition for data-mechanism fusion in high-dimensional dynamical systems

Abstract

To overcome the high online correction cost of full-rank Physics-Constrained Dynamic Mode Decomposition (PCDMD) in high-dimensional dynamical systems, this paper proposes a low-rank PCDMD method. This method reformulates predictive evolution, covariance propagation, and physics-based correction in a unified low-rank space. It introduces a prediction basis and a correction basis to reduce the accuracy loss caused by low-rank approximation. The method is evaluated on mathematical PDE benchmarks and complex cases, including catalytic fixed-bed reactor, lid-driven cavity flow, and flow past a square cylinder. The results show that low-rank PCDMD preserves high prediction accuracy while reducing online cost by more than 20-fold in representative cases. The method remains effective under noisy data, sparse observations, and non-uniform temporal sampling. The proposed method therefore provides a practical route for extending data-mechanism fusion models to complex high-dimensional dynamical systems.