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.

翻译：泛函微分方程在数学物理的众多领域扮演着基础性角色，包括流体动力学（霍普特征泛函方程）、量子场论（施温格-戴森方程）以及统计物理。尽管其重要性显著，但在数学物理中，求解泛函微分方程的计算方法长期面临挑战。本文通过引入新的逼近理论和高性能计算算法，致力于在张量流形上求解泛函微分方程。我们的方法首先使用高维偏微分方程来逼近泛函微分方程，随后利用高性能并行张量算法在低秩张量流形上求解此类高维偏微分方程。通过将其应用于Burgers-Hopf泛函微分方程（该方程描述了从随机初始状态演化的Burgers方程随机解的特征泛函），验证了所提方法的有效性。