We investigate quantum-inspired tensor networks (QTNs) for approximating flow maps of hydrodynamic partial differential equations (PDEs). Motivated by the effective low-rank structure that emerges after tensorization of discretized transport and diffusion dynamics, we encode PDE states as matrix product states (MPS) and represent the evolution operator as a structured low-rank matrix product operator (MPO) in tensor-train form (e.g., arising from finite-difference discretizations assembled in MPO form). The MPO is applied directly in MPS form, and rank growth is controlled via canonicalization and SVD-based truncation after each step. We provide theoretical context through standard matrix product properties, including exact MPS representability bounds, local optimality of SVD truncation, and a Lipschitz-type multi-step error propagation estimate. Experiments on one- and two-dimensional linear advection-diffusion and nonlinear viscous Burgers equations demonstrate accurate short-horizon prediction, favorable scaling in smooth diffusive regimes, and error growth in nonlinear multi-step predictions.
Pseudospectral method for solving PDEs using Matrix Product States
Institute of Fundamental Physics IFF-CSIC（基础物理研究所）；Millennium Institute for Research in Optics（光学研究千年研究所）；Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción（康塞普西翁大学物理与数学科学学院）
Friedrichs' systems discretized with the Discontinuous Galerkin method: domain decomposable model order reduction and Graph Neural Networks approximating vanishing viscosity solutions
A quantum-inspired multi-level tensor-train monolithic space-time method for nonlinear PDEs
Khalifa University of Science and Technology（哈利法科学技术大学）；Technology Innovation Institute（技术创新研究所）；Emirates Nuclear Energy Company（阿联酋核能公司）；Emirates Nuclear Technology Center（阿联酋核技术中心）
Space-time deep neural network approximations for high-dimensional partial differential equations
Faculty of Mathematics, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院数学学院）；Seminar for Applied Mathematics, Department of Mathematics, ETH Zurich（苏黎世联邦理工学院数学系应用数学研讨班）；Department for Applied Mathematics, University of Freiburg（弗莱堡大学应用数学系）
Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore（数学科学系，物理与数学科学学院，南洋理工大学，新加坡）；Faculty of Mathematics, Bielefeld University, Germany（数学学院，比勒菲尔德大学，德国）
Modeling Unknown Nonlocal PDE Systems via Flow Map Learning
Department of Mathematics, The Ohio State University（俄亥俄州立大学数学系）；Department of Mathematics and Statistics, Missouri University of Science and Technology（密苏里科技大学数学与统计系）
