Tensor Networks Offer Scalable Solution for 2D Fluid Dynamics Simulations

A tensor network solver for two-dimensional Euler equations achieved acceptance in the final phase of the Airbus-BWM Group Challenge 2024. The method scales polylogarithmically with mesh size, offering a potential solution to the computational demands of high-resolution fluid dynamics simulations.

The accurate simulation of fluid dynamics remains a substantial computational challenge, particularly when dealing with compressible flows such as those encountered in aerospace engineering. Direct numerical simulation, while theoretically precise, often suffers from limitations imposed by the ‘curse of dimensionality’ – the exponential increase in computational resources required as the complexity of the simulation increases. Researchers are now exploring alternative approaches, including quantum-inspired algorithms, to mitigate these challenges. A new technical report details one such solver, utilising tensor networks to achieve a polylogarithmic scaling with mesh size – a significant improvement in both runtime and memory requirements. The work is presented by Raghavendra Dheeraj Peddinti, Stefano Pisoni, Egor Tiunov, Alessandro Marini, and Leandro Aolita, from institutions including the Technology Innovation Institute and Hamburg University of Technology, in their report entitled ‘Technical report on a quantum-inspired solver for simulating compressible flows’.

Matrix Product States Offer Novel Approach to Fluid Dynamics

Research demonstrates the successful application of Matrix Product States (MPS) – a mathematical technique originating in condensed matter physics – to the simulation of fluid dynamics. This represents a departure from conventional Computational Fluid Dynamics (CFD) methods and offers potential advantages in computational efficiency and scalability.

The research team implemented an MPS solver for both one-dimensional (1D) and two-dimensional (2D) Euler equations – a set of partial differential equations governing the motion of inviscid fluids. A key feature of this approach is its inherent data compression. Simulations achieved compression rates of 98.7% for velocity fields, 55% for density, and 47% for pressure, substantially reducing memory requirements and computational load.

Critically, the solver exhibits polylogarithmic scaling with mesh size. Traditional CFD methods typically scale polynomially, meaning computational cost increases rapidly with increasing resolution. Polylogarithmic scaling implies a significantly slower increase in computational demand as the simulation becomes more detailed, potentially circumventing the ‘curse of dimensionality’ that limits the scope of conventional simulations.

The solver’s accuracy was validated against the well-established Sod shock tube problem – a standard benchmark for evaluating CFD methods. Results demonstrated the solver’s ability to reproduce shock wave formation and propagation accurately. The implementation has been extended to the 2D Euler equations, broadening its applicability. Furthermore, the work has been accepted for participation in the Airbus-BMW Challenge 2024, indicating its relevance to industrial applications.

The demonstrated data compression and polylogarithmic scaling offer the potential for substantial reductions in computational cost for complex fluid dynamics simulations. This could enable the modelling of larger and more detailed systems than currently feasible, with implications for diverse fields including aerospace engineering, automotive design, and environmental modelling.

This research presents a promising alternative to traditional CFD methods, leveraging techniques from condensed matter physics to address limitations in computational cost and scalability, potentially unlocking new possibilities for accurate and efficient modelling of complex fluid flows.