Dynamic tensor approximation of high-dimensional nonlinear PDEs

TL;DR

This paper introduces a novel dynamic tensor approximation method for efficiently solving high-dimensional nonlinear PDEs by projecting solutions onto low-rank tensor manifolds, enabling accurate and scalable computations.

Contribution

It develops a new approach combining functional tensor decomposition with dynamic approximation to solve high-dimensional PDEs efficiently.

Findings

Successfully applied to a 4D Fokker-Planck equation

Accurately predicts relaxation to equilibrium

Maintains low-rank tensor representations during evolution

Abstract

We present a new method based on functional tensor decomposition and dynamic tensor approximation to compute the solution of a high-dimensional time-dependent nonlinear partial differential equation (PDE). The idea of dynamic approximation is to project the time derivative of the PDE solution onto the tangent space of a low-rank functional tensor manifold at each time. Such a projection can be computed by minimizing a convex energy functional over the tangent space. This minimization problem yields the unique optimal velocity vector that allows us to integrate the PDE forward in time on a tensor manifold of constant rank. In the case of initial/boundary value problems defined in real separable Hilbert spaces, this procedure yields evolution equations for the tensor modes in the form of a coupled system of one-dimensional time-dependent PDEs. We apply the dynamic tensor approximation to a four-dimensional Fokker-Planck equation with non-constant drift and diffusion coefficients, and demonstrate its accuracy in predicting relaxation to statistical equilibrium.