Quantum-Inspired Tensor Networks for Approximating PDE Flow Maps

TL;DR

This paper explores quantum-inspired tensor networks to efficiently approximate PDE flow maps, leveraging low-rank structures and tensor decompositions for improved computational modeling of hydrodynamic equations.

Contribution

It introduces a novel approach using matrix product states and operators to represent PDE dynamics, with theoretical analysis and practical experiments demonstrating effectiveness.

Findings

Accurate short-horizon predictions for PDEs.

Favorable scaling in diffusive regimes.

Controlled rank growth via tensor truncation.

Abstract

We investigate quantum-inspired tensor networks (QTNs) for approximating flow maps of hydrodynamic partial differential equations (PDEs). Motivated by the effective low-rank structure that emerges after tensorization of discretized transport and diffusion dynamics, we encode PDE states as matrix product states (MPS) and represent the evolution operator as a structured low-rank matrix product operator (MPO) in tensor-train form (e.g., arising from finite-difference discretizations assembled in MPO form). The MPO is applied directly in MPS form, and rank growth is controlled via canonicalization and SVD-based truncation after each step. We provide theoretical context through standard matrix product properties, including exact MPS representability bounds, local optimality of SVD truncation, and a Lipschitz-type multi-step error propagation estimate. Experiments on one- and two-dimensional linear advection-diffusion and nonlinear viscous Burgers equations demonstrate accurate short-horizon prediction, favorable scaling in smooth diffusive regimes, and error growth in nonlinear multi-step predictions.