Quantum-inspired technique simulates turbulence with high speed
Quantum-inspired “tensor networks” can simulate the behaviour of turbulent fluids in just a few hours rather than the several days required for a classical algorithm. The new technique, developed by physicists in the UK, Germany and the US, could advance our understanding of turbulence, which has been called one of the greatest unsolved problems of classical physics.
Turbulence is all around us, found in weather patterns, water flowing from a tap or a river and in many astrophysical phenomena. It is also important for many industrial processes. However, the way in which turbulence arises and then sustains itself is still not understood, despite the seemingly simple and deterministic physical laws governing it.
The reason for this is that turbulence is characterized by large numbers of eddies and swirls of differing shapes and sizes that interact in chaotic and unpredictable ways across a wide range of spatial and temporal scales. Such fluctuations are difficult to simulate accurately, even using powerful supercomputers, because doing so requires solving sets of coupled partial differential equations on very fine grids.
An alternative is to treat turbulence in a probabilistic way. In this case, the properties of the flow are defined as random variables that are distributed according to mathematical relationships called joint Fokker-Planck probability density functions. These functions are neither chaotic nor multiscale, so they are straightforward to derive. However, they are nevertheless challenging to solve because of the high number of dimensions contained in turbulent flows.
For this reason, the probability density function approach was widely considered to be computationally infeasible. In response, researchers turned to indirect Monte Carlo algorithms to perform probabilistic turbulence simulations. However, while this approach has chalked up some notable successes, it can be slow to yield results.
Highly compressed “tensor networks”
To overcome this problem, a team led by Nikita Gourianov of the University of Oxford, UK, decided to encode turbulence probability density functions as highly compressed “tensor networks” rather than simulating the fluctuations themselves. Such networks have already been used to simulate otherwise intractable quantum systems like superconductors, ferromagnets and quantum computers, they say.
These quantum-inspired tensor networks represent the turbulence probability distributions in a hyper-compressed format, which then allows them to be simulated. By simulating the probability distributions directly, the researchers can then extract important parameters, such as lift and drag, that describe turbulent flow.
Importantly, the new technique allows an ordinary single CPU (central processing unit) core to compute a turbulent flow in just a few hours, compared to several days using a classical algorithm on a supercomputer.
This significantly improved way of simulating turbulence could be particularly useful in the area of chemically reactive flows in areas such as combustion, says Gourianov. “Our work also opens up the possibility of probabilistic simulations for all kinds of chaotic systems, including weather or perhaps even the stock markets,” he adds.
The researchers now plan to apply tensor networks to deep learning, a form of machine learning that uses artificial neural networks. “Neural networks are famously over-parameterized and there are several publications showing that they can be compressed by orders of magnitude in size simply by representing their layers as tensor networks,” Gourianov tells Physics World.
The study is detailed in Science Advances.
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