Tensor Network Computation: Representations, Algebra, and Applications

现代数据收集和计算模拟产生了许多高维数据集。这样的高维数据集代表了计算数学中的关键挑战之一。众所周知，主要的困难在于维度的诅咒，即以传统方式表示和分析高维函数所需的自由度数量随着维度的数量呈指数增长。最近，张量网络作为一种很有前途的工具出现在理论和计算物理界，用于表示和处理高维函数和概率。本项目对张量网络方法进行了系统的计算研究。这项调查是为处理高维数据集提供一般框架的第一步。该项目集中在张量网络的四个方面。(1)在张量网络骨架算法的基础上，针对非齐次系统和一般张量网络提出了有效的张量压缩算法，并进行了并行实现。(2)研究人员计划开发新的算法来构建张量网络，无论是通过采样还是根据现有的但冗余的张量表示。该方法将从随机化数值代数、非线性优化和多尺度方法中汲取工具和思想。(3)该项目将开发用于处理张量网络表示中的高维函数的基本代数运算的算法。这些基本运算包括加法、减法、逐项乘法、逐项求逆、逐项函数应用，以及张量网络算子形式的线性运算符的应用。(4)最后一个目标是研究张量网络在传统统计和量子力学领域之外的新应用，例如在数值齐化、超高维不确定性量化以及控制和分子动力学中的高维偏微分方程组。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Modern data collection and computational simulations produce many high-dimensional data sets. Such high-dimensional data sets represent one of the key challenges in computational mathematics. As is well-known, the main difficulty lies in the curse of the dimensionality, that is, the number of degrees of freedom required to represent and analyze high-dimensional functions in the traditional way grows exponentially with the number of dimensions. Recently, tensor networks have emerged, mostly from the theoretical and computational physics community, as a promising tool for representing and manipulating high-dimensional functions and probabilities. This project undertakes a systematic computational study of the tensor network approach. The investigation serves as an initial step in providing a general framework for work with high-dimensional data sets. The project focuses on four aspects of tensor networks. (1) The investigator aims to improve on recent work in tensor network skeletonization, where the main task is to develop efficient tensor contraction algorithms for inhomogeneous systems and general tensor networks, with an emphasis on parallel implementation. (2) The investigator plans to develop novel algorithms for constructing a tensor network either via sampling or from an existing but redundant tensor representation. The approach will draw tools and ideas from randomized numerical algebra, nonlinear optimization, and multiscale methods. (3) The project will develop algorithms for basic algebraic operations for manipulating high-dimensional functions in the tensor network representation. These basic operations include addition, subtraction, entry-wise multiplication, entry-wise inversion, entry-wise function application, and application of linear operators in tensor network operator form. (4) The last objective is to investigate new applications of tensor networks outside the traditional realm of statistical and quantum mechanics, for example in numerical homogenization, uncertainty quantification in very high dimension, and high-dimensional partial differential equations in control and molecular dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.