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Tensor Networks in Fluid Dynamics for Cost Reduction

1. The Problem: Challenges in Fluid Dynamics

Fluid dynamics, especially turbulent flow modeling, is extremely computationally expensive due to:

  • Multiscale nature: Turbulence involves eddies of various sizes interacting with each other.
  • High-dimensional equations: The state of a turbulent flow is described by the Navier-Stokes equations, which are nonlinear partial differential equations (PDEs).
  • Exponential complexity: To accurately resolve turbulence, traditional methods need ( M^d ) grid points, where:
    • ( M ) is the number of discretization points per dimension.
    • ( d ) is the number of dimensions (velocity, position, time, chemical composition, etc.).
    • In practical cases, ( d ) can be 1000 or more, making simulations intractable.

Traditional Methods in Fluid Dynamics

  1. Direct Numerical Simulation (DNS)

    • Solves the Navier-Stokes equations directly but is computationally infeasible for high Reynolds numbers.
    • Scales as ( O(M^d) ) in memory and computation.

  2. Large Eddy Simulation (LES)

    • Simulates only large turbulence scales, modeling small ones.
    • Still computationally expensive and requires subgrid-scale models.

  3. Monte Carlo (MC) Methods

    • Uses random sampling to estimate turbulence properties.
    • Slow convergence, requiring many samples for accuracy.

Tensor Network Approach

Instead of simulating every detail of turbulence, the method in the paper compresses turbulence probability distributions into a tensor network representation, drastically reducing complexity.

2. The Solution: Tensor Networks for Fluid Dynamics

2.1. Reformulating Fluid Dynamics as a Probability Distribution

  • Instead of tracking exact fluid velocities and pressures, we model the probability density function (PDF) of flow variables.

  • The Fokker-Planck equation describes the evolution of these PDFs:

    ∂f∂t+⟨Ui⟩∂f∂xi−∂∂xi((γ+γSGS)∂f∂xi)=∂∂ϕα(Ωmix(ϕα−⟨Φα⟩)f)−∂∂ϕα(Sαf)\frac{\partial f}{\partial t} + \langle U_i \rangle \frac{\partial f}{\partial x_i} - \frac{\partial}{\partial x_i} \left( (\gamma + \gamma_{SGS}) \frac{\partial f}{\partial x_i} \right) = \frac{\partial}{\partial \phi_{\alpha}} \left( \Omega_{mix} (\phi_{\alpha} - \langle \Phi_{\alpha} \rangle) f \right) - \frac{\partial}{\partial \phi_{\alpha}} \left( S_{\alpha} f \right)

    where:

    • ( f(x,t) ) = joint probability density function of turbulence.
    • ( U_i(x,t) ) = velocity components.
    • ( \phi_{\alpha} ) = mass fraction of chemical species.
    • ( \gamma, \gamma_{SGS} ) = diffusion coefficients (molecular and subgrid-scale).
    • ( S_{\alpha} ) = reaction terms.

  • The curse of dimensionality makes solving this PDE directly infeasible using standard finite-difference (FD) methods.

2.2. How Tensor Networks Solve This

Tensor networks (TNs) allow a compressed representation of turbulence PDFs, avoiding the need to store the entire ( M^d ) grid.

  • Instead of storing ( M^d ) points, we approximate the PDF using a Matrix Product State (MPS) tensor network:
f(ϕ1,ϕ2,x,t)=∑{i}Ai1[1]Ai2[2]...AiN[N] f(\phi_1, \phi_2, x, t) = \sum_{\{i\}} A^{[1]}_{i_1} A^{[2]}_{i_2} ... A^{[N]}_{i_N}
  • ( A^{[n]}_{i_n} ) are tensors representing interactions between different variables.
  • The compression is controlled by the bond dimension ( \chi ).
  • Instead of needing ( M^d ) storage, TNs only require ( O(d \log M) ) storage.

Key advantages:

  1. Computational Cost Reduction

    • Finite-difference schemes scale as ( O(M^d) ).
    • Tensor networks scale as ( O(d \log M) ).
    • This means memory and computational cost reduction by factors of ( O(10^6) ) and ( O(10^3) ), respectively.

  2. High Accuracy Without Monte Carlo Noise

    • Unlike Monte Carlo methods, tensor networks directly solve equations without randomness.
    • Faster convergence.

  3. Runs on Classical Computers

    • Unlike quantum computing algorithms, this method only requires classical CPUs.
    • Feasible for industrial and research applications without a supercomputer.

3. Proof of Efficiency in Fluid Dynamics

The study demonstrates the power of this method through simulations of reactive turbulent flows, where two chemical species mix and react.

3.1. Benchmark Against Traditional Methods

  • Test Case: 5D turbulence PDF simulation of a chemically reactive jet.
  • Computational Cost Comparison:
    • Finite Difference (FD) Scheme: Would require ( 128^5 = 33.6 ) billion grid points.
    • Tensor Network (MPS) Scheme: Uses only ( O(10^5) ) parameters, a ( 10^6 ) reduction in memory.
    • Time Complexity: Reduced from ( O(M^d) ) to ( O(d \log M) ).

3.2. Accuracy and Convergence

  • The root mean square error (RMSE) analysis shows that the MPS method converges polynomially with bond dimension ( \chi ).
  • Even at low bond dimensions (e.g., ( \chi = 32 )), the MPS simulation provides results indistinguishable from DNS.

3.3. Example Application: Jet Mixing

  • A turbulent jet with chemical reactions was simulated.
  • The MPS representation correctly captured:
    • Turbulent mixing behavior.
    • Chemical species distributions.
    • Probability density evolution.

Sources & citation

Tensor networks enable the calculation of turbulence probability distributions: https://www.science.org/doi/10.1126/sciadv.ads5990