A novel tensor-based modal decomposition method for reduced order modeling and optimal sparse sensor placement

Data-driven modal decomposition methods have become fundamental algorithms in constructing reduced order models (ROMs), which serve as a critical tool in the digital twin modeling of large complex systems. Classical modal decomposition methods are particularly adept at handling matrix-form input data that contains columnized snapshots without retaining spatial structure and neighborhood information. Focusing on addressing this problem, this paper proposed a Tensor-Based Modal Decomposition (TBMD) method to deal with the high-order tensor-form input data. TBMD method is effective in finding the potential low-order manifolds of a given high-order dynamical system with low energy loss within fewer decomposited modes. Moreover, TBMD is extended to be used in the field of optimal sensor placement via the development of a tensor-based QR factorization method and a tensor-based compressive sensing algorithm. The tensor-based QR factorization method is able to consider the spatial structure and neighborhood information of the basis matrices data in tensor form. This is the first time that a compressive sensing model has been discussed and solved for the frontal slice-wise product of a tensor and a vector. Subsequently, comprehensive case studies are conducted based on the cylinder wake dataset, airfoil vortex dataset, sea surface temperature dataset, eigenfaces dataset and turbulent channel flow dataset. The experiment results show that TBMD can accurately capture lower-order manifolds, and the tensor-based QR factorization is effective in optimizing sensor placement with fewer sensors while maintaining high reconstruction accuracy. The proposed methods in this paper is promising to be utilized in the construction of large-scale complex system digital twins, in which TBMD can serve as a fundamental algorithm in constructing ROM.