Tensor approximation of functional differential equations

Abstract

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.

Figures (8)

• Figure 1: Examples of tensor networks. The vertices represent tensor modes (functions) used in the decomposition. The edges connecting to vertices represent summation over an index (contraction) between two modes. The free edges represent input variables $x_j$ ($j=1,\ldots,n$).
• Figure 2: Sketch of implicit and explicit step-truncation integration methods. Given a tensor $\bm f_k$ with multilinear rank $\bm s$ on the tensor manifold ${\cal M}_{\bm s}$, we first perform an explicit time-step with any conventional time-stepping scheme. The explicit step-truncation integrator then projects $\bm \Psi_{\Delta t}(\bm G,\bm f_k)$ to a tensor manifold with rank $\bm s$. The multilinear rank $\bm s$ is chosen adaptively based on desired accuracy and stability constraints rodgers2020step-truncation. On the other hand, the implicit step-truncation method takes $\bm \Psi_{\Delta t}(\bm G,\bm f_k)$ as input and generates a sequence of fixed-point iterates ${\bm f}^{[j]}$ shown as dots connected with blue lines. The last iterate is then projected onto a low rank tensor manifold, illustrated here also as a red line landing on ${\cal M}_{\bm s}$. This operation is equivalent to the compression step in the HT/TT-GMRES algorithm described in dolgov2013ttgmres.
• Figure 3: Sketch of the logical flow to transform a random nonlinear initial value problem (IVP) for the Burgers equation into the (linear) Burgers-Hopf FDE, which is linear. The Burgers-Hopf FDE is subsequently approximated using the methods outlined in Section \ref{['sec:FTE_to_hyperPDEs']}. This yields an $N$-dimensional characteristic function equation of the form \ref{['hyper1']}. By taking the inverse Fourier transform of such characteristic function equation and integrating it with respect to the phase variable we obtain the $N$-dimensional CDF equation shown above. The CDF equation is then solved using the proposed step-truncation tensor methods.
• Figure 4: Local truncation errors (calculated as $O(\Delta t^{p+1})$ for and order $p$ method) of the proposed step-truncation tensor methods. These errors are computed by comparing one time step of step-truncation method with its Richardson extrapolation. It is seen that explicit/implicit Euler and explicit/implicit midpoint step-truncation methods have accuracy of order one and order two, respectively.
• Figure 5: Two joint CDF of $u(0,t)$ and $u(\pi,t)$. The CDF is computed by generating numerical solutions to \ref{['eqn:cdf-burgers']} for $N=20$ using the proposed step-truncation tensor methods, and then marginalizing the solution in the remaining 18 variables. We also show a Monte-Carlo estimate of the joint CDF obtained by sampling $5\times 10^6$ solutions to \ref{['eqn:burgers-rxn']}.