Tensor Network Fluid Simulations in Structured Domains Using the Lattice Boltzmann Method

Lukas Gross; Elie Mounzer; David M. Wawrzyniak; Josef M. Winter; Nikolaus A. Adams

Paper

Tensor Network Fluid Simulations in Structured Domains Using the Lattice Boltzmann Method

Abstract

High-fidelity fluid simulations are central to understanding transport phenomena, yet resolving large or geometrically complex systems remains computationally prohibitive with existing methods. Recently, methods based on tensor-networks, commonly known as a quantum-inspired approach, were proposed to efficiently simulate flow in simple domains, where complexity emerges mostly from turbulence. Here, we substantially extend the understanding of such methods by demonstrating for the first time that also flows governed by translational or approximate symmetries of the geometry exhibit very low effective complexity in matrix product state (MPS) form. To this end, we introduce a tensor-network formulation of the lattice Boltzmann method based on MPS and demonstrate the generality of the method on three-dimensional flows through structured media and complex vascular geometries, establishing that tensor-network techniques can efficiently resolve fluid dynamics in complex domains previously inaccessible to MPS approaches. We show that in structured media, MPS representation yields compression ratios exceeding two orders of magnitude while preserving physical structure and dynamical fidelity. This reduction enables systematic numerical exploration of regimes that were previously intractable. Our results position tensor networks as a scalable paradigm for continuum mechanics.