AI 中文总结
研究将有限元与张量网络连接用于解析算子方程，通过有限元离散化诱导多线性相互作用张量层次结构，转化方程为加权残差优化问题，实现一维线性和非线性扩散计算并重现传统解，为算子方程求解提供通用变分语言和基础。
AI 中文摘要
算子方程在科学与工程的定量建模中起着支撑作用。有限元方法将连续算子方程离散为有限维代数系统，而张量网络为相关离散系统提供灵活的变分表示。本文开发了一个将有限元与张量网络连接起来用于解析算子方程的框架。该方法的强大之处在于能将高度非线性偏微分方程转化为线性矩阵方程。通过有限元离散化可诱导出多线性相互作用张量层次结构，进而在共同代数结构中表达微分、积分、非线性、记忆和延迟方程。所得系统被重新表述为张量网络自由度上的加权残差优化问题。一维线性和非线性扩散的矩阵乘积态计算在保持连续性和诺伊曼边界条件的同时，以可控误差重现传统解。该框架为解析算子方程提供了通用变分语言，建立了有限元数值形式与张量网络变分算法的直接联系，为基于张量网络和量子启发的算子方程求解方法奠定了基础。
英文摘要
Operator equations (OEs) underpin quantitative modeling across science and engineering. Finite-element (FE) methods discretize continuous OEs into finite-dimensional algebraic systems, whereas tensor networks (TNs) provide flexible variational representations of correlated discrete systems. Here, we develop a framework that connects FE with TN for analytic OEs. The power of this method comes from its ability to convert highly non-linear partial differential equations into linear matrix equations. In particular, we show that FE discretization induces a hierarchy of multilinear interaction tensors, through which differential, integral, nonlinear, memory, and delay equations can be expressed within a common algebraic structure. The resulting systems are reformulated as weighted-residual optimization problems over TN degrees of freedom. Matrix-product-state calculations for one-dimensional linear and nonlinear diffusion reproduce conventional solutions with controlled error while preserving continuity and Neumann boundary conditions. The framework provides a common variational language for analytic OEs and establishes a direct connection between FE numerical formalism and TN variational algorithms, offering a general foundation for TN-based and quantum-inspired approaches to solving OEs.
Comments11 pages, 8 figures
