  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.
