Tensor Network Formulation of the Lattice Boltzmann Method

Our approach maps the discretized LBM state space into TT format, decomposing velocity distribution functions into a chain of core tensors whose bounded rank controls the compression ratio. In this TT decomposition, the linear streaming step reduces to an exact low-rank TT matrix. For the collision step, we adopt a Multiple Relaxation Time (MRT) operator to perform the necessary transformations between velocity and moment spaces, using efficient element-wise operations that allow relaxation of the distribution functions in momentum space. Furthermore, we allow the computation for complex geometries using binary masks. After each arithmetic operation, a truncation of the TT ranks must be performed. We support different compression strategies, namely standard singular value decomposition, alternating least squares, and tensor cross-interpolation.

Our TT-LBM solver is implemented using an in-house TT library built on PyTorch and directly integrated into our classical LBM solver. We validate our TT-LBM algorithm against the classical reference implementation across several benchmark cases, including Taylor-Green Vortex (TGV) decay and the lid-driven cavity flow at various Reynolds numbers. We compare different TT compression methods and rank bounds in terms of accuracy, compression ratio, and computational cost. The results demonstrate that the TT decomposition reproduces the reference solutions with high fidelity while achieving significant memory compression, confirming a first step toward the practical viability of the approach for realistic engineering flow problems.

The practical value of the TT-MRT-LBM solver lies in its ability to circumvent the volumetric memory limits that constrain high-resolution fluid simulations. We show that, for realistic flows with bounded tensor ranks, the method enables simulations on grids that would otherwise exceed the memory of current HPC systems. This opens a path toward fully resolved, industrial-scale fluid dynamics problems on existing hardware, not through algorithmic shortcuts, but through a fundamentally more compact representation of the simulation data. Insights from our results regarding the low-rank construction of the TT matrix and elementwise operations might help develop native quantum computing algorithms for the LBM and quantum CFD.