发表机构
MIT（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于纠缠基张量网络的PDE求解器，通过增广系数空间和MPS表示实现多项式规模计算，并扩展到时间依赖问题。
AI 中文摘要
我们开发了一种基于纠缠基表示的偏微分方程（PDE）求解器的有限元方法（FEM），该表示在我们的配套工作中引入。通过将非线性有限元方程提升到增广系数空间，控制PDE连同边界、初始和单元间约束可以统一表示为二次残差最小化问题。尽管这个增广空间随单元数量呈指数增长，但其张量积结构使其能够使用张量网络高效表示。以矩阵乘积态（MPS）作为具体示例，我们展示了密度矩阵重整化群（DMRG）扫描能够实现逐单元优化，而无需显式构造完整的增广空间。对于有界键维，所得计算成本随有限元数量呈多项式增长。我们通过隐式时间离散化将该框架扩展到时间相关问题，并使用扩散方程证明了在网格和多项式细化下的收敛性。
英文摘要
We develop a finite element method (FEM) for partial differential equation (PDE) solver based on the entanglement-basis representation introduced in our companion work. By lifting non-linear finite-element equations into an augmented coefficient space, the governing PDE together with boundary, initial, and inter-element constraints can be expressed through a unified quadratic residual minimization. Although this augmented space grows exponentially with the number of elements, its tensor-product structure allows it to be represented efficiently using tensor networks. Using the matrix product state (MPS) as a concrete example, we show that density matrix renormalization group (DMRG) sweeps enable element-by-element optimization without explicitly constructing the full augmented space. For bounded bond dimension, the resulting computational cost scales polynomially with the number of finite elements. We extend the framework to time-dependent problems through implicit temporal discretization and demonstrate convergence under both mesh and polynomial refinement using diffusion equations.
Comments15 pages, 13 figures
