Turbulent Flows Simulated with 99.99% Accuracy Using Matrix Product States

Turbulent reactive flows present a longstanding challenge in combustion modelling, requiring accurate prediction of how turbulence influences chemical reactions and vice versa, a task complicated by the vast range of physical scales involved. Robert Pinkston, Nikita Gourianov, and Hirad Alipanah, all from the University of Pittsburgh, alongside colleagues including Dieter Jaksch from the University of Oxford, now present a novel approach using matrix product states, a technique originally developed in condensed matter physics. Their method offers a viable alternative to traditional direct numerical simulation, achieving significant memory reductions, up to 99. 99% for certain variables based on analysis of existing data, while accurately capturing key flow physics such as the effects of heat release and the formation of complex shock structures. This advance, demonstrated through simulations of shear flow with chemical reactions, promises to unlock the ability to model increasingly complex turbulent combustion systems, overcoming limitations imposed by computational resources.

Tensor Networks Compress Fluid Dynamics Simulations Researchers are exploring the use of tensor networks, a powerful mathematical tool originating in quantum physics, to dramatically improve computational fluid dynamics (CFD) simulations, particularly those involving turbulence and chemical reactions. This work addresses the fundamental challenge in CFD: the immense computational resources required to accurately model complex interactions across a wide range of scales. By employing tensor networks as a compressed representation of flow solutions, scientists aim to reduce memory requirements and accelerate simulations without sacrificing accuracy. The central idea involves representing fluid flow variables, such as velocity, temperature, and species concentrations, using matrix product states, a specific type of tensor network. This approach leverages the fact that many fluid flow solutions exhibit low entanglement, meaning information can be efficiently encoded in a compressed format. Researchers are also investigating how quantum algorithms and quantum computing hardware can further enhance CFD simulations, exploring techniques like variational quantum algorithms and direct quantum simulation of partial differential equations. This growing field focuses heavily on turbulence, a notoriously difficult phenomenon to simulate due to its multiscale nature. The ability to compress solutions using tensor networks is seen as a key enabler for both classical and quantum CFD, allowing scientists to tackle increasingly complex problems.

Matrix Product State Simulation of Turbulent Flows Scientists have developed a novel computational methodology based on matrix product states (MPS), a form of tensor network, to simulate turbulent reacting flows and address the significant computational demands of resolving complex interactions between turbulence and chemistry. This work departs from traditional direct numerical simulation (DNS) by employing the MPS framework, initially successful in many-body quantum physics, to efficiently represent flow states and compress the required memory. The study focuses on a two-dimensional reacting shear flow governed by six coupled partial differential equations, encoding and time-evolving the entire system within the MPS framework. Crucially, the methodology involves truncating all transport variables, velocity, temperature, species concentrations, and their spatial derivatives, throughout the simulation, enabling substantial data compression. Researchers implemented a specialized algorithm to evolve the system forward in time using the MPS representation, accurately capturing the complex interplay between fluid dynamics and chemical reactions.

The team assessed the impact of compressibility and heat release on the truncated simulation, demonstrating the method’s ability to handle physically realistic conditions. Experiments demonstrate a 30% reduction in memory requirements for all transport variables while maintaining excellent agreement with results obtained from conventional DNS. A priori analysis of data at higher Reynolds numbers reveals the potential for even greater compression, reaching levels as high as 99. 99% for certain variables, suggesting the MPS approach could unlock simulations of far more complex turbulent combustion systems.

Matrix Product States Compress Turbulence Simulations Scientists have developed a new methodology, based on matrix product states (MPS), as an alternative to direct numerical simulation (DNS) for modeling turbulent reactive flows. This work addresses the challenge of accurately predicting how turbulence affects chemical reaction rates and how chemistry, in turn, influences fluid dynamics, a key problem in combustion modeling. The MPS approach represents a significant advancement by efficiently compressing the vast amount of memory required for high-resolution simulations. Experiments demonstrate that the MPS method achieves memory reductions of 30% for all transport variables while maintaining excellent agreement with results obtained from traditional DNS. This compression is achieved through a tensor network framework, originally developed in condensed matter physics, which allows for an exponential reduction in memory usage compared to conventional methods. The MPS accurately captures complex flow physics, including the reduction of mixing caused by heat release and compressibility, and the formation of eddy shocklets at high Mach numbers. Further analysis of DNS data at higher Reynolds numbers reveals even more substantial compression, with some transport variables compressed by as much as 99. 99%. This remarkable level of compression suggests the MPS method is highly effective at capturing the essential features of turbulent flows with significantly reduced computational cost. The researchers validated the MPS approach by simulating a temporally developing jet (TDJ), a flow characterized by a shear layer and the formation of coherent vortices. The simulations were conducted on a 128×128 grid, and statistical data were obtained by sampling along the streamwise direction. The degree of mixing was quantified using the vorticity thickness, and the flow dynamics were characterized by Reynolds shear stresses. This innovative approach promises to enable simulations of complex turbulent combustion systems that were previously intractable due to limitations in computational resources.

Matrix Product States Simulate Turbulent Combustion Flows This work presents a new computational method, based on matrix product states, for simulating turbulent reacting flows, a challenging area of research in combustion modeling.

The team successfully developed an algorithm that accurately replicates results from direct numerical simulation, but with significantly reduced computational demands. Specifically, the method achieves up to 30% memory reduction while faithfully capturing key flow physics, including the impact of compressibility and heat release on mixing, and the formation of eddy shocklets at high speeds. Analysis of the method’s performance indicates that its accuracy improves as the size of the simulated system increases, with truncation error decreasing as the system grows. Furthermore, the researchers demonstrate the potential for substantial compression ratios, ranging from 10−5 to 10−1, which could dramatically reduce the computational resources needed for simulating complex combustion processes involving numerous chemical species and reactions. Future work will focus on adapting the algorithm to further reduce computational demands per timestep, potentially enhancing its accuracy and efficiency. This advancement offers a promising pathway towards more efficient and detailed simulations of turbulent reacting flows, ultimately aiding the understanding and optimization of combustion processes. 👉 More information 🗞 Matrix Product State Simulation of Reacting Shear Flows 🧠 ArXiv: https://arxiv.org/abs/2512.13661 Tags: