Tensor approximation of functional differential equations

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 equations), 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 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.