AI 中文总结
该研究针对对流-扩散-反应偏微分方程，提出量子启发张量网络框架，将离散解场编码为MPS，微分算子表示为MPOs，用显式欧拉更新进行时间积分。通过与高精度解比较，验证其在多维度问题上保持紧凑、稳定、准确，展现了张量网络用于PDE模拟的潜力。
AI 中文摘要
我们提出了一个用于求解对流-扩散-反应（ADR）偏微分方程的量子启发张量网络框架。离散化的解场被编码为矩阵乘积态（MPS），而微分算子被表示为矩阵乘积算子（MPOs）。时间积分完全以张量网络形式使用具有可控截断的显式欧拉更新进行。该方法在一维和二维ADR问题上进行评估，并与高精度龙格-库塔参考解进行比较。数值结果表明，所提出的表示在一系列动态区域中保持紧凑、稳定和准确。求解器在整个模拟过程中保持较小的键维度的同时捕获局部解轮廓和全局可观测量。这些结果突出了张量网络作为多空间维度中PDE模拟的有效结构保持工具的潜力。
英文摘要
We present a quantum-inspired tensor-network framework for solving advection-diffusion-reaction (ADR) partial differential equations. Discretized solution fields are encoded as matrix product states (MPS), while differential operators are represented as matrix product operators (MPOs). Time integration is performed entirely in tensor-network form using explicit Euler updates with controlled truncation. The method is evaluated on one- and two-dimensional ADR problems and compared with high-accuracy Runge-Kutta reference solutions. Numerical results show that the proposed representation remains compact, stable, and accurate across a range of dynamical regimes. The solver captures both local solution profiles and global observables while maintaining small bond dimensions throughout the simulation. These results highlight the potential of tensor networks as efficient structure-preserving tools for PDE simulation in multiple spatial dimensions.
Comments6 pages, 7 figures
