Exploring Low-Rank Structures in Shallow Water Equations Using Quantum Tensor Networks

The shallow water equations capture essential features of many geophysical flows, including nonlinearity and geostrophic balance, making them a valuable model for testing novel computational frameworks. In this work, we investigate their low-rank structure using quantum-inspired tensor network methods, with a focus on matrix product states (MPS). By evolving the shallow water fields in MPS form, we track the bond dimension—analogous to quantum entanglement—to quantify solution complexity over time. Despite a transient growth, we observe saturation in the bond dimension, suggesting that the system exhibits emergent low-rank behavior. We further assess how grid resolution influences the efficiency and accuracy of MPS compression. These findings suggest that quantum tensor networks not only facilitate efficient representations of multiscale geophysical dynamics but also provide new insights into flow structure and nonlinearity.