报告人：黄记祖

报告题目：

Derivatives of tree tensor networks and its applications in Runge–Kutta methods

时间：2025年5月7日10:00—11:00

地点：数学楼2-1会议室

摘要：

Tree tensor networks (TTNs) provide a compact and structured representation of high-dimensional data, making them valuable in various areas of computational mathematics and physics. In this work, we present a rigorous mathematical framework for expressing high-order derivatives of functional TTNs, both with or without constraints. Our framework decomposes the total derivative of a given TTN into a summation of TTNs, each corresponding to the partial derivatives of the original TTN. Using this decomposition, we derive the Taylor expansion of vector-valued functions subject to ordinary differential equation constraints or algebraic constraints imposed by Runge--Kutta (RK) methods. As a concrete application, we employ this framework to construct order conditions for RK methods. Due to the intrinsic tensor properties of partial derivatives and the separable tensor structure in RK methods, the Taylor expansion of numerical solutions can be obtained in a manner analogous to that of exact solutions using tensor operators. This enables the order conditions of RK methods to be established by directly comparing the Taylor expansions of the exact and numerical solutions, eliminating the need for mathematical induction. For a given function f, we derive sharper order conditions that go beyond the classical ones, enabling the identification of situations where a standard RK scheme of order p achieves unexpectedly higher convergence order for the particular function. These results establish new connections between tensor network theory and classical numerical methods, potentially opening new avenues for both analytical exploration and practical computation.

报告人简介：

黄记祖, 现任中国科学院数学与系统科学研究院副研究员。2004年毕业于湖南大学，获学士学位。2012年在中国科学院数学与系统科学研究院获理学博士学位（硕博连读），导师：曹礼群研究员。2012年7月到2014年7月，在中国科学院软件研究所从事博士后研究工作。2014年7月，入职中国科学院数学与系统科学研究院，主要研究领域为多尺度模型和并行算法。

邀请人：王飞 教授