标题： 泛函微分方程的张量逼近 显示英文标题

标题： Tensor approximation of functional differential equations

摘要： 泛函微分方程（FDEs）在数学物理的许多领域中起着基础性作用，包括流体动力学（Hopf特征泛函方程）、量子场论（Schwinger-Dyson方程）和统计物理。 尽管它们的重要性，计算FDEs的解仍然是数学物理中的一个长期挑战。 在本文中，我们通过引入新的近似理论和高性能计算算法来解决这一挑战，这些算法专门用于在张量流形上求解FDEs。 我们的方法涉及使用高维偏微分方程（PDEs）来近似FDEs，然后利用高性能并行张量算法在低秩张量流形上求解这些高维PDEs。 所提出的方法的有效性通过其在Burgers-Hopf FDE上的应用得到验证，该方程描述了从随机初始状态演化的Burgers方程的随机解的特征泛函。 显示英文摘要

摘要： Functional Differential Equations (FDEs) play a fundamental role in many areas of mathematical physics, including fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equation), and statistical physics. Despite their significance, computing solutions to FDEs remains a longstanding challenge in mathematical physics. In this paper we address this challenge by introducing new approximation theory and high-performance computational algorithms designed for solving FDEs on tensor manifolds. Our approach involves approximating FDEs using high-dimensional partial differential equations (PDEs), and then solving such high-dimensional PDEs on a low-rank tensor manifold leveraging high-performance parallel tensor algorithms. The effectiveness of the proposed approach is demonstrated through its application to the Burgers-Hopf FDE, which governs the characteristic functional of the stochastic solution to the Burgers equation evolving from a random initial state.