DiffVortex: A quantum-inspired fast solver for incompressible flows
Abstract
Matrix-product-state (MPS) and quantized tensor-train (QTT) representations offer a promising framework for computational fluid dynamics, with logarithmic run time and memory scaling in mesh size when bond dimensions remain bounded. Existing MPS-based solvers are Eulerian: they compress velocity fields and enforce immersed-body boundary conditions through static masks or body-fitted coordinates. Moving or flexible bodies complicate this approach: deformation requires updating MPS masks every time step, while filamentary vorticity in high-Reynolds-number flows may resist low-rank compression. We propose a complementary approach: compress the Biot--Savart kernel rather than the flow state, evolve vorticity using Lagrangian vortex particles, and represent immersed bodies by bound vortex sheets on Lagrangian surface points. Our main theoretical result is a lifting theorem: QTT rank r of the kernel implies matrix-product-operator (MPO) rank at most 2d r for the corresponding convolution operator in d dimensions, independent of grid resolution. This bound is sharp for our interleaved ordering. With quantics tensor cross interpolation, the moment-to-local stage of a single-level fast multipole method becomes a precomputed tensor-network contraction with quasi-linear complexity, retaining Lagrangian flexibility for immersed bodies. Our PyTorch implementation (\texttt{DiffVortex}) is end-to-end differentiable, runs on GPUs, and handles 6.7×107 particles on a single device. We validate against reference vortex-method results for an impulsively started cylinder and exact and reference solutions for inviscid elliptical vortices. We also demonstrate two-dimensional flows past rigidly moving, multiple, and deforming bodies, including gradients propagated through the full time integration.
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BibTeX
@misc{loo2026diffvortex,
title = {{DiffVortex: A quantum-inspired fast solver for incompressible flows}},
author = {Yu Jun Loo and Silas Alben and Shravan Veerapaneni},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30012},
eprint = {2610.30012},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30012},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - DiffVortex: A quantum-inspired fast solver for incompressible flows
AU - Yu Jun Loo
AU - Silas Alben
AU - Shravan Veerapaneni
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30012
UR - https://arxiv.org/abs/2610.30012
AB - Matrix-product-state (MPS) and quantized tensor-train (QTT) representations offer a promising framework for computational fluid dynamics, with logarithmic run time and memory scaling in mesh size when bond dimensions remain bounded. Existing MPS-based solvers are Eulerian: they compress velocity fields and enforce immersed-body boundary conditions through static masks or body-fitted coordinates. Moving or flexible bodies complicate this approach: deformation requires updating MPS masks every time step, while filamentary vorticity in high-Reynolds-number flows may resist low-rank compression. We propose a complementary approach: compress the Biot--Savart kernel rather than the flow state, evolve vorticity using Lagrangian vortex particles, and represent immersed bodies by bound vortex sheets on Lagrangian surface points. Our main theoretical result is a lifting theorem: QTT rank r of the kernel implies matrix-product-operator (MPO) rank at most 2^d r for the corresponding convolution operator in d dimensions, independent of grid resolution. This bound is sharp for our interleaved ordering. With quantics tensor cross interpolation, the moment-to-local stage of a single-level fast multipole method becomes a precomputed tensor-network contraction with quasi-linear complexity, retaining Lagrangian flexibility for immersed bodies. Our PyTorch implementation (\texttt{DiffVortex}) is end-to-end differentiable, runs on GPUs, and handles 6.7×10^7 particles on a single device. We validate against reference vortex-method results for an impulsively started cylinder and exact and reference solutions for inviscid elliptical vortices. We also demonstrate two-dimensional flows past rigidly moving, multiple, and deforming bodies, including gradients propagated through the full time integration.
ER -