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arXiv:2410.19747

Tensor Train Multiplication

Alexios A. Michailidis, Christian Fenton, Martin Kiffner

Published Oct 10, 2024• 0 citations
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Abstract

We present the Tensor Train Multiplication (TTM) algorithm for the elementwise multiplication of two tensor trains with bond dimension $χ$. The computational complexity and memory requirements of the TTM algorithm scale as $χ^3$ and $χ^2$, respectively. This represents a significant improvement compared with the conventional approach, where the computational complexity scales as $χ^4$ and memory requirements scale as $χ^3$.We benchmark the TTM algorithm using flows obtained from artificial turbulence generation and numerically demonstrate its improved runtime and memory scaling compared with the conventional approach. The TTM algorithm paves the way towards GPU accelerated tensor network simulations of computational fluid dynamics problems with large bond dimensions due to its dramatic improvement in memory scaling.

DOI
10.48550/arxiv.2410.19747

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Tensor network algorithms can efficiently simulate complex quantum many-body systems by utilizing knowledge of their structure and entanglement. These methodologies have been adapted recently for solving the Navier-Stokes equations, which describe a spectrum of fluid phenomena, from the aerodynamics of vehicles to weather patterns. Within this quantum-inspired paradigm, velocity is encoded as matrix product states (MPS), effectively harnessing the analogy between interscale correlations of fluid dynamics and entanglement in quantum many-body physics. This particular tensor structure is also called quantics tensor train (QTT). By utilizing NVIDIA's cuQuantum library to perform parallel tensor computations on GPUs, our adaptation speeds up simulations by up to 12.1 times. This allows us to study the algorithm in terms of its applicability, scalability, and performance. By simulating two qualitatively different but commonly encountered 2D flow problems at high Reynolds numbers up to 1 × 10 7 using a fourth-order time stepping scheme, we find that the algorithm has a potential advantage over direct numerical simulations in the turbulent regime as the requirements for grid resolution increase drastically. In addition, we derive the scaling χ = O ( poly ( 1 / ε ) ) for the maximum bond dimension χ of MPS representing turbulent flow fields, with an error ε , based on the spectral distribution of turbulent kinetic energy. Our findings motivate further exploration of related quantum algorithms and other tensor network methods.
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The Hadamard product of two tensors in the tensor-train (TT) format is a fundamental operation across various applications, such as TT-based function multiplication for nonlinear differential equations or convolutions. However, conventional methods for computing this product typically scale as at least O(χ4)\mathcal{O}(χ^4)O(χ4) with respect to the TT bond dimension (TT-rank) χχχ, creating a severe computational bottleneck in practice. By combining randomized tensor-train sketching with slice selection via interpolative decomposition, we introduce Recursive Sketched Interpolation (RSI), a ``scale product'' algorithm that computes the Hadamard product of TTs at a computational cost of O(χ3)\mathcal{O}(χ^3)O(χ3). Benchmarks across various TT scenarios demonstrate that RSI offers superior scalability compared to traditional methods while maintaining comparable accuracy. We generalize RSI to compute more complex operations, including Hadamard products of multiple TTs and other element-wise nonlinear mappings, without increasing the complexity beyond O(χ3)\mathcal{O}(χ^3)O(χ3).
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