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Mean Field Games for Controlling Coherent Structures in Nonlinear Fluid Systems

Yuan Gao, Di Qi

TL;DR

The paper tackles the problem of steering coherent structures in nonlinear turbulent flows by marrying mean-field game theory with fluid dynamics. It develops two interlinked MFG formulations: MFG-1 for controlling tracer density under a given flow and MFG-2 for direct vorticity control, both expressed through functional Hamilton–Jacobi equations that couple forward density evolution with backward value functions. A novel fixed-point iterative strategy decouples the nonlinear MFG-2 system by leveraging a closely related, easier decoupled MFG-1, enabling fast convergence across 1D and 2D prototype tests based on a modified Burgers equation. The results demonstrate robust, efficient control of multi-scale turbulent transport and lay groundwork for extending to higher-dimensional, more complex flows with potential broad scientific and engineering impact.

Abstract

This paper discusses the control of coherent structures in turbulent flows, which has broad applications among complex systems in science and technology. Mean field games have been proved a powerful tool and are proposed here to control the stochastic Lagrangian tracers as players tracking the flow field. We derive optimal control solutions for general nonlinear fluid systems using mean field game models, and develop computational algorithms to efficiently solve the resulting coupled forward and backward mean field system. A precise link is established for the control of Lagrangian tracers and the scalar vorticity field based on the functional Hamilton-Jacobi equations derived from the mean field models. New iterative numerical strategy is then constructed to compute the optimal solution with fast convergence. We verify the skill of the mean field control models and illustrate their practical efficiency on a prototype model modified from the viscous Burger's equation under various cost functions in both deterministic and stochastic formulations. The good model performance implies potential effectiveness of the strategy for more general high-dimensional turbulent systems.

Mean Field Games for Controlling Coherent Structures in Nonlinear Fluid Systems

TL;DR

The paper tackles the problem of steering coherent structures in nonlinear turbulent flows by marrying mean-field game theory with fluid dynamics. It develops two interlinked MFG formulations: MFG-1 for controlling tracer density under a given flow and MFG-2 for direct vorticity control, both expressed through functional Hamilton–Jacobi equations that couple forward density evolution with backward value functions. A novel fixed-point iterative strategy decouples the nonlinear MFG-2 system by leveraging a closely related, easier decoupled MFG-1, enabling fast convergence across 1D and 2D prototype tests based on a modified Burgers equation. The results demonstrate robust, efficient control of multi-scale turbulent transport and lay groundwork for extending to higher-dimensional, more complex flows with potential broad scientific and engineering impact.

Abstract

This paper discusses the control of coherent structures in turbulent flows, which has broad applications among complex systems in science and technology. Mean field games have been proved a powerful tool and are proposed here to control the stochastic Lagrangian tracers as players tracking the flow field. We derive optimal control solutions for general nonlinear fluid systems using mean field game models, and develop computational algorithms to efficiently solve the resulting coupled forward and backward mean field system. A precise link is established for the control of Lagrangian tracers and the scalar vorticity field based on the functional Hamilton-Jacobi equations derived from the mean field models. New iterative numerical strategy is then constructed to compute the optimal solution with fast convergence. We verify the skill of the mean field control models and illustrate their practical efficiency on a prototype model modified from the viscous Burger's equation under various cost functions in both deterministic and stochastic formulations. The good model performance implies potential effectiveness of the strategy for more general high-dimensional turbulent systems.
Paper Structure (26 sections, 8 theorems, 70 equations, 10 figures, 1 table, 3 algorithms)

This paper contains 26 sections, 8 theorems, 70 equations, 10 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction and background
  2. Formulation of the fluid control equations
  3. Governing equation for the transport of a scalar field
  4. Stochastic Lagrangian equations for tracer transport
  5. Control of the fluid fields with mean field games
  6. Control of the tracer density field with a given fluid solution
  7. Control of the vorticity flow field as a potential mean field game
  8. SDE formulations for the mean field game models
  9. Functional Hamilton-Jacobi equations for the value functions
  10. Functional HJE for the MFG-1 model
  11. Functional HJE for the MFG-2 model
  12. A link between the MFG-1 and MFG-2 models
  13. Computational strategies for the MFG models
  14. Practical choices of the cost functions
  15. Algorithms to solve the decoupled MFG-1 model
  16. ...and 11 more sections

Key Result

Proposition 1

Given the flow vorticity field $q_s\left(x\right),s\in \left[t, T\right]$, the optimal tracer density $\rho_{s}\left(x\right)$ under the cost function eq:loss_tracer1 with $L\left(\alpha\right)=\frac{1}{2}\vert\alpha\vert^2$ is given by the solution of the following MFG-1 system The corresponding optimal control can be solved by $\alpha_{s}\left(x\right) = \nabla \phi_{s}\left(x\right)$.

Figures (10)

  • Figure 6.1: An illustration of the steady solution of the MVB equation with $q_t=-\partial_x u_t$.
  • Figure 6.2: Optimal controlled solutions of MFG-1 model for tracer transport with different loss functions.
  • Figure 6.3: Sampling non-Gaussian PDFs by controlling the tracer density in the SDE model.
  • Figure 6.4: Improvement in the target value function $\mathcal{I}\left(q^{\mu}_s,\alpha^{\mu}_s\right)-\mathcal{I}\left(q^{n}_s,\alpha^{n}_s\right)$ with different values of $\mu$ during the first two iterations using $L_2$ and KL-divergence cost.
  • Figure 6.5: Total value function $\mathcal{I}^{n}=\mathcal{I}(q_{s}^{\left(n\right)},\alpha_{s}^{\left(n\right)})$ and the errors in the updated states $\lbrace q_{s}^{\left(n\right)},\alpha_{s}^{\left(n\right)}\rbrace$ during the updating iterations using both $L_{2}$ and KLD loss.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Corollary 3
  • Corollary 4
  • Proposition 5
  • proof
  • ...and 8 more