Fundamental study of tensor network-based finite difference methods in computational fluid dynamics

Recently, quantum-inspired tensor network (TN) methods have been introduced to efficiently reduce the computational cost of flow simulations. In the TN methods proposed by previous studies, fluid variables are represented as matrix product states (MPS) constructed by singular value decomposition (SVD). These approaches have demonstrated that the TN methods can significantly reduce the number of parameters required to represent direct numerical simulation results, thereby improving computational efficiency. Although these findings highlight the potential of TN methods for flow computations, their fundamental properties remain insufficiently understood. This study aims to clarify the basic characteristics of TN-based finite difference methods in computational fluid dynamics. Previous studies [1, 2] proposed two types of MPS representations. A numerical simulation of two-dimensional sine wave advection is conducted to elucidate the fundamental characteristics of those MPS representations. The analysis reveals that efficient data compression cannot be achieved unless an MPS representation well-suited to the structure of flow fields is employed. Furthermore, this study investigates the relation between SVD-based compression and truncation errors due to finite difference discretization.

1. Gourianov et al., Nature Computational Science, 2(1) (2022) 30-37.

2. Kiffner and Jaksch, Physical Review Fluids, 8(12) (2023) 124101.