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 $2^{d}r$ 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 $6.7\times 10^7$ 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.