Tensor approximation of functional differential equations
<|reference_start|>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 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.<|reference_end|>
@article{rodgers2024tensor,
    title={Tensor approximation of functional differential equations},
    author={Abram Rodgers and Daniele Venturi},
    journal={arXiv preprint arXiv:2403.04946},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.04946},
    primaryClass={math.NA cs.NA math-ph math.MP physics.comp-ph}
}