Convexifying Mean-Field Control: An Occupation-Measure and Frank-Wolfe Approach

Chaoying Pei; Di Yu; Sixiong You

REVIEW 4 major objections 3 minor 21 references

Convexifying Mean-Field Control: An Occupation-Measure and Frank-Wolfe Approach

T0 review · 4 major / 3 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Mean-field control of a swarm can be reformulated as a convex optimization over occupation measures, and each Frank–Wolfe iteration reduces to a standard optimal-control solve.

desk verdict Genuinely new combination and clean convex calculus, but the central theorem is an explicitly incomplete proof sketch and the O(1/k) rate is imported from self-citation; worth a serious referee. read the letter →

arxiv 2607.22678 v1 pith:R76G5BOX submitted 2026-07-10 math.OC stat.ME

Di Yu , Sixiong You , Chaoying Pei This is my paper

classification math.OCstat.ME MSC 49J4590C25

keywords mean-fieldcontroloccupationmeasuresFrank–Wolfeconvexoptimizationswarmroboticsoptimalmeasure-valuedsafetyconstraints

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that mean-field control—the problem of steering the distribution of a large swarm of agents—can be lifted into a convex optimization over occupation measures, rather than being solved through PDEs on a grid or through approximation-heavy machine learning. The central move is to encode every swarm behavior as a pair of measures (running and terminal) and to show the nonconvex interaction cost becomes a quadratic form that is convex whenever the interaction kernel is positive semidefinite. On this measure space, a Frank–Wolfe algorithm runs at O(1/k) convergence, and each linear subproblem reduces to solving a single classical optimal-control problem with a stage cost updated from the current measure. If the reduction holds, a principled and discretization-free route opens to large-scale swarm coordination with safety and interaction constraints.

What carries the argument

The central object is the occupation-measure pair (µ, ν): µ records the time-state-control distribution of an agent's running trajectory and ν records its terminal state, with the Liouville identity (6) encoding the dynamics as linear constraints. The carrying mechanism is the first variation g_µ in equation (12), which makes the Frank–Wolfe subproblem a linear functional over ∆; Theorem 4 reduces that linear program to a classical optimal control problem for one trajectory, so each outer iteration is an optimal-control solve rather than a grid-based update.

What would settle it

Construct a feasible occupation measure (µ, ν) satisfying the Liouville identity whose support cannot be generated by any single classical trajectory (for instance, with a nonconvex velocity set f(x,U)), then evaluate the linear objective ⟨g_µ, µ⟩ + ⟨Ψ, ν⟩ at that measure and compare it to the minimum value over all classical trajectories; if the latter is strictly larger, the equality in Theorem 4 fails and the O(1/k) argument loses its oracle.

Watch

Extended reading notes

Core claim

The paper's key claim is that problem (P)—minimizing the sum of individual running cost, a quadratic interaction term, and terminal cost over a set ∆ of occupation measures constrained by a linear Liouville equation—is a convex formulation of the mean-field control problem. Theorem 2 shows J is convex when the interaction kernel W is positive semidefinite, and ∆ is convex and weak-* compact, so a solution exists. Theorem 3 computes the first variation as g_µ = ℓ0 + 2λ ∫ W(x−y)δ(t−t′) dµ, which is affine in the measure. Theorem 4 then states that the Frank–Wolfe linear oracle over ∆ is minimized by a single admissible trajectory, turning each iteration into a classical optimal control problem

Load-bearing premise

The load-bearing premise is that every feasible occupation measure in ∆ can be decomposed as a mixture of classical trajectories, so that the linear Frank–Wolfe subproblem is minimized by a single classical trajectory; Theorem 4's proof leaves this as a sketch, deferring relaxed controls to an omitted approximation argument.

Editorial extensions

If this is right

If the reduction is correct, mean-field control inherits the O(1/k) convergence guarantee of Frank–Wolfe without any state-space discretization, with computational effort shifted to the per-iteration optimal-control subproblem.
The convex formulation supports interaction and safety constraints directly in measure form (repulsion kernels, obstacle potentials), so the same algorithm handles them without modification.
The algorithm outputs a weighted collection of admissible trajectories, which can be viewed as a plan-level distribution that later can be sampled to drive a finite swarm.
Numerical evidence suggests practical use: 3D scenarios with multiple obstacles solve in minutes on a workstation, where grid-based PDE solvers are impractical.
The method opens the possibility of tackling MFC problems with a broad class of existing optimal-control solvers as black-box oracles.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One step beyond the paper: because the FW subproblem is a classical optimal control problem, the framework could be wrapped inside a model-predictive-control loop, re-solving the convex program as the swarm's measured distribution deviates from the plan.
The proof gap around relaxed controls suggests the convexification may not be exact for nonconvex velocity sets; a natural test would be to run the algorithm on a system with a nonconvex control set and compare the FW gap against the best single-trajectory lower bound.
If the convex measure program is exact, it provides an alternative to learned swarm policies with certificates, and one could even use the FW objective value as a stopping criterion for safe online deployment.
The occupation-measure format may extend to mean-field games with potential structure, since the same linear oracle reduction appears in a symmetric game setting.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The new thing here is real: lifting mean-field control into occupation measures and running Frank-Wolfe directly in that infinite-dimensional space, with each iteration collapsing to a classical optimal control problem. The convexity argument and the first-variation computation are clean and standard, and the numerical work—at least what's shown—is a plausible first demonstration. The 2D agreement with a PDE baseline is encouraging, and the 3D obstacle run suggests the method can scale to settings where grids struggle.

The soft spots are exactly where the reader's report focuses. Theorem 4, the bridge that justifies the whole algorithm, is a proof sketch with an explicit unproved assumption: every feasible occupation measure is a distribution over classical trajectories. The authors write that a full proof needs relaxed controls and approximation arguments, 'omitted here for brevity.' That isn't a minor gap—it's the load-bearing step. Without a density or compactness argument, the reduction to a single optimal control trajectory isn't justified, and the Frank-Wolfe oracle could fail. The O(1/k) rate is likewise not proved here; it's imported from the authors' own prior work via citation. That's acceptable if the cited results really cover this setting, but this setting is new enough that a statement of the transferred conditions is needed. There's also an initial-condition mismatch: the experiments use a Gaussian initial distribution while the theory assumes a single initial condition. The authors acknowledge this in Remark 1, but it means the numerical validation doesn't directly exercise the stated problem.

None of this kills the core idea. The convex reformulation is sound, and the first-variation formalism is correct. The gaps are fixable—complete the relaxed-control proof, state the convergence theorem with hypotheses, and reconcile the initial-condition issue—but as written the paper's central claim is not yet established. This is exactly the kind of submission that should go to a serious referee, with the expectation of heavy revision. I wouldn't cite it in my own work until the proof is completed.

Referee Report

4 major / 3 minor

Summary. The paper proposes a mean-field control (MFC) formulation for large swarms, lifted into the space of occupation measures. The main object is the infinite-dimensional convex program (P) over measure pairs (µ,ν) satisfying a Liouville identity, with a bilinear interaction term. The authors show convexity of the objective and feasible set, compute the first variation, and design a Frank-Wolfe algorithm (Algorithm 1) whose linear subproblem is claimed to reduce to a classical optimal control problem. Convergence at O(1/k) is asserted by reference to prior FW-on-measure-space work. Numerical experiments in 2D and 3D with obstacle potentials and Gaussian repulsion are reported to validate the approach.

Significance. If fully established, the paper would give a principled, discretization-free route to MFC that retains convexity and provable rates, while reducing each iteration to a standard optimal control oracle. The convexity proof and first-variation computation are clean and self-contained, and the numerical experiments demonstrate potential scalability. However, the central algorithmic claim rests on Theorem 4, whose proof sketch explicitly omits the rigorous relaxed-control/approximation argument. The O(1/k) convergence guarantee is also not derived in the manuscript but imported from self-cited prior work without verifying its hypotheses here. These gaps are load-bearing for the method's validity.

major comments (4)

[§IV, Theorem 4 and Eq. (15)] The reduction of the FW linear subproblem to a single classical optimal control trajectory is the core of Algorithm 1, but the proof is only a sketch and explicitly assumes that decomposed paths admit classical control representations. For general nonlinear f and nonconvex velocity set f(x,U), the extreme points of the relaxed-control occupation-measure set are not necessarily classical arcs. A rigorous proof requires either a classical-representability theorem (e.g., convexity of f(x,U) or an affine-in-u structure) or a relaxed-control treatment plus an approximation argument. As written, Eq. (15) and Algorithm 1 Step 3 lack justification.
[§IV, convergence rate (after Eq. (15))] The claim that γ_k = 2/(k+2) yields O(1/k) convergence is not proved in this paper; it is referred to [13],[14]. Standard Frank-Wolfe O(1/k) bounds require additional structure beyond convexity and compactness—e.g., Lipschitz-continuous gradient or bounded curvature of J on Δ. The manuscript does not verify any such condition for the interaction term in (P). Please state the precise theorem being invoked and confirm that its hypotheses hold for this infinite-dimensional objective.
[§V-A, numerical setup] The model in Section II assumes x_i(0)=x_0 for all agents, so ρ_0=δ_{x0}. The 2D verification, however, specifies an initial distribution ρ_0=N((0.12,0.12), σ_0^2 I) with σ_0=0.07. If this is generated by sampling non-identical initial states, the theory as presented does not cover it; if it is a non-atomic distribution used only for visualization, the text is misleading. This inconsistency should be resolved by either extending the formulation to general initial measures or aligning the experiment with the common-initial-state assumption.
[§V-A, subproblem solver claim] The text states that the inner optimal control subproblem 'reduces to a quadratic program (QP) with linear dynamics and quadratic cost.' With a Gaussian interaction kernel W, the term 2λ∑ α_i W(x-x_i) in g_µ is nonconvex, so the subproblem is not a QP. A lightweight gradient-based routine may only find a local minimum, while Frank-Wolfe convergence requires global optimality of the linear minimization oracle. This discrepancy should be clarified; if global optimality is not guaranteed, the numerical validation does not test the algorithm as analyzed.

minor comments (3)

[Eq. (5) and Eq. (P)] The interaction term uses δ(t−t′) as a formal Dirac distribution on the product measure. Since µ need not be absolutely continuous in both time copies, the expression should be defined precisely, e.g., as ∫_0^T ∫∫ W(x−y) dµ_t(x,u) dµ_t(y,v) dt, using the disintegration of µ with respect to time.
[Theorem 2] The assumption that W is positive semidefinite is introduced in the proof but not stated as a hypothesis in the problem formulation (P). It should be stated explicitly, together with the required measurability/integrability conditions beyond boundedness.
[Notation in Section II] The paper alternates between µ_N, μ_N, and similar notation. For readability, use a single consistent symbol for the averaged occupation measure and reserve subscripts for iteration indices.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation chain; the only caveats are a proof-sketch gap in Theorem 4 and a minor self-citation for the FW convergence rate.

full rationale

The paper's central claims are not circular in the sense of a result reducing to its own inputs by construction. Theorem 1 derives the occupation-measure form of the N-agent cost by direct measure identities (7)-(9), and the averaged Liouville identity (6) follows from the fundamental theorem of calculus along each admissible trajectory. Theorem 2 proves convexity by an explicit expansion, and Theorem 3 computes the first variation by direct differentiation. The numerical experiments are benchmarked against an independent PDE solution in 2D, so the main reformulation is externally validated rather than self-referential. The main concerns are two. First, Theorem 4's proof of the FW subproblem reduction is explicitly a sketch: it assumes 'these decomposed paths admit classical control representations' and states that 'a fully rigorous proof requires addressing relaxed controls and approximation arguments [21], which are omitted here for brevity.' This is a genuine rigor gap in the load-bearing oracle step, but it is an unproved assumption, not a definitional or fitted-input circularity. Second, the O(1/k) convergence guarantee is imported by the sentence 'We refer to [13], [14] for detailed descriptions and convergence analysis of FW methods in the measure-optimization setting,' where [13]-[15] are self-citations by the first author. That is a minor reliance on self-citation rather than a circular reduction, and the cited works are published or in-principle checkable FW-in-measure-space results. Overall, the central convex formulation is self-contained; the score reflects only these minor non-circular weaknesses.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The mathematical derivation introduces no fitted parameters; all free parameters belong to the numerical illustrations. The central claim rests on standard occupation-measure and superposition results, but the classical-control reduction and FW convergence are assumed or self-cited rather than derived here.

free parameters (5)

interaction kernel amplitude kappa (lambda_W) = 1.5 (2D), 25 (3D)
Chosen by hand; in Section V, kappa is set so that repulsion at spacing d_min balances the terminal pull. Not fitted to data, but ad hoc for the illustrations.
interaction kernel width sigma_W = 0.15 (2D)
Sensing radius chosen by hand in Section V-A.
obstacle potential weight beta = 10^3
Barrier strength chosen by hand so that obstacles are avoided.
cost weights alpha, lambda_Psi = alpha=0.1, lambda_Psi=30
Hand-picked in Section V; these define the demonstration, not the framework.
initial standard deviation sigma_0 = 0.07 (2D)
Width of the Gaussian initial distribution in Section V-A, which is not representable under the identical-initial-condition assumption of Section II-A.

assumptions (4)

domain assumption All N agents share the same initial condition x_i(0)=x0, reducing the initial empirical distribution to delta_{x0}.
Stated in Section II-A; it is load-bearing for the Liouville constraint in (10) and is contradicted by the 2D experiment's Gaussian rho_0 in Section V-A.
domain assumption X and U are compact and f, l0, Psi, W have enough regularity (bounded continuous W PSD, finite integrals) for weak-* compactness and existence.
Used in Section III for compactness and convexity and in Corollary 1; exact regularity conditions on f are not stated.
domain assumption Every (mu,nu) in Delta admits a probabilistic representation as a distribution over admissible trajectories, and the linear subproblem's minimum is attained by a single classical trajectory.
Used in Theorem 4 proof sketch; the paper explicitly defers the rigorous relaxed-control argument to reference [21].
domain assumption Frank-Wolfe on the infinite-dimensional measure set Delta converges at O(1/k) with step sizes gamma_k=2/(k+2).
Asserted in Section IV by reference to the authors' own prior papers [13],[14]; no theorem or conditions are stated in this manuscript.

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Cite this review

Pith. "Pith review of Convexifying Mean-Field Control: An Occupation-Measure and Frank-Wolfe Approach." pith.science (2026). https://pith.science/paper/R76G5BOX

@misc{pith2026260722678,
  author       = {Pith},
  title        = {Pith review of: Convexifying Mean-Field Control: An Occupation-Measure and Frank-Wolfe Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R76G5BOX}},
  note         = {Machine review of arXiv:2607.22678}
}

read the original abstract

Large-scale robotic swarms motivate the use of mean-field control (MFC). Classical partial differential equation (PDE)-based formulations provide a principled framework but can become computationally challenging in higher dimensions, whereas machine learning achieves scalability at the cost of approximation and guarantees. In this work, we establish an optimization-based framework that lifts the MFC problem into the space of occupation measures, resulting in a convex relaxation formulated as an optimization over measures. The resulting problem is solved using a Frank-Wolfe (FW) algorithm in the measure space, with each iteration reduced to a tractable optimal control problem. This approach retains the O(1/k) convergence rate of FW, avoids discretization of the state space, and naturally incorporates interaction and safety constraints. Numerical experiments demonstrate agreement with analytic and PDE-based baselines in two dimensions and show that the method scales to three-dimensional environments with multiple obstacles, where standard grid-based PDE solvers become impractical. A full 3D instance with ten obstacles is solved in minutes on a standard workstation, underscoring the practicality and scalability of the proposed framework.

Figures

Figures reproduced from arXiv: 2607.22678 by the authors.

**Figure 1.** compares the FW solution, visualized as time slices of the occupation measures µk (blue points, with darker points indicating larger weights), with the PDE benchmark density (orange heatmap) in the presence of a repulsive potential and a single obstacle (dashed circle). Both approaches yield similar swarm evolution and terminal distributions, with the agents coordinating to avoid the obstacle and converge to the ta… view at source ↗

**Figure 2.** Objective J: FW (blue) approaches PDE (orange). FW objective decreasing over iterations and approaching the PDE benchmark value. This demonstrates convergence in objective value and validates that the FW scheme faithfully approximates the PDE solution in this more challenging setting. B. 3D Swarm with Repulsion and Multiple Obstacles: Scalability In this part, we solve a 3D swarm control problem with ten obstacles … view at source ↗

**Figure 3.** 3D swarm with repulsion and 10 obstacles. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

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Reference graph

Works this paper leans on

21 extracted references · 1 linked inside Pith

[13]

Deterministic and stochastic frank-wolfe recursion on probability spaces,

D. Yu, S. G. Henderson, and R. Pasupathy, “Deterministic and stochastic frank-wolfe recursion on probability spaces,”Mathematics of Operations Research, 2025

2025
[14]

Frank-wolfe recursions for the emergency response problem on measure spaces,

D. Yu, S. G. Henderson, and R. Pasupathy, “Frank-wolfe recursions for the emergency response problem on measure spaces,”arXiv preprint arXiv:2507.09808, 2025

arXiv 2025
[1]

Mean-field models in swarm robotics: A survey,

K. Elamvazhuthi and S. Berman, “Mean-field models in swarm robotics: A survey,”Bioinspiration & Biomimetics, vol. 15, no. 1, p. 015001, 2019

2019
[2]

Mean-field optimal control,

M. Fornasier and F. Solombrino, “Mean-field optimal control,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 20, no. 4, pp. 1123–1152, 2014

2014
[3]

Continuum swarm tracking control: A geometric perspective in wasserstein space,

M. Emerick and B. Bamieh, “Continuum swarm tracking control: A geometric perspective in wasserstein space,” in2023 62nd IEEE Conference on Decision and Control (CDC), pp. 1367–1374, IEEE, 2023

2023
[4]

Mean field deep reinforcement learning for fair and efficient uav control,

D. Chen, Q. Qi, Z. Zhuang, J. Wang, J. Liao, and Z. Han, “Mean field deep reinforcement learning for fair and efficient uav control,”IEEE Internet of Things Journal, vol. 8, no. 2, pp. 813–828, 2020

2020
[5]

A machine learning framework for solving high-dimensional mean field game and mean field control problems,

L. Ruthotto, S. J. Osher, W. Li, L. Nurbekyan, and S. W. Fung, “A machine learning framework for solving high-dimensional mean field game and mean field control problems,”Proceedings of the National Academy of Sciences, vol. 117, no. 17, pp. 9183–9193, 2020

2020
[6]

Mean-field control barrier functions: A framework for real-time swarm control,

S. W. Fung and L. Nurbekyan, “Mean-field control barrier functions: A framework for real-time swarm control,” in2025 American Control Conference (ACC), pp. 1–7, IEEE, 2025

2025

Show all 21 references

[7]

Kernel expansions for high-dimensional mean-field control with non-local interactions,

A. Vidal, S. W. Fung, S. Osher, L. Tenorio, and L. Nurbekyan, “Kernel expansions for high-dimensional mean-field control with non-local interactions,” in2025 American Control Conference (ACC), pp. 4164– 4171, IEEE, 2025

2025
[8]

Learning in mean field games: the fictitious play,

P. Cardaliaguet and S. Hadikhanloo, “Learning in mean field games: the fictitious play,”ESAIM: Control, Optimisation and Calculus of Variations, vol. 23, no. 2, pp. 569–591, 2017

2017
[9]

Nonlinear optimal control via occupation measures and lmi-relaxations,

J. B. Lasserre, D. Henrion, C. Prieur, and E. Tr ´elat, “Nonlinear optimal control via occupation measures and lmi-relaxations,”SIAM journal on control and optimization, vol. 47, no. 4, pp. 1643–1666, 2008

2008
[10]

Optimum experimental designs V, with applications to sys- tematic and rotatable designs,

J. Kiefer, “Optimum experimental designs V, with applications to sys- tematic and rotatable designs,” inProceedings of the fourth Berkeley symposium on mathematical statistics and probability, vol. 1, pp. 381– 405, Univ of California Press, 1960

1960
[11]

The alternating descent con- ditional gradient method for sparse inverse problems,

N. Boyd, G. Schiebinger, and B. Recht, “The alternating descent con- ditional gradient method for sparse inverse problems,”SIAM Journal on Optimization, vol. 27, no. 2, 2017

2017
[12]

A mean field view of the landscape of two-layer neural networks,

S. Mei, A. Montanari, and P.-M. Nguyen, “A mean field view of the landscape of two-layer neural networks,”Proceedings of the National Academy of Sciences, vol. 115, jul 2018

2018
[15]

The derivative-free fully-corrective frank-wolfe algorithm for optimizing functionals over probability spaces,

D. Yu, S. G. Henderson, and R. Pasupathy, “The derivative-free fully-corrective frank-wolfe algorithm for optimizing functionals over probability spaces,” in2025 Winter Simulation Conference (WSC), pp. 3358–3369, IEEE, 2025

2025
[16]

Generalized conditional gradient and learning in potential mean field games,

P. Lavigne and L. Pfeiffer, “Generalized conditional gradient and learning in potential mean field games,”Applied Mathematics & Optimization, vol. 88, no. 3, p. 89, 2023

2023
[17]

Carmona, F

R. Carmona, F. Delarue,et al.,Probabilistic theory of mean field games with applications I-II, vol. 3. Springer, 2018

2018
[18]

Large- scale nonconvex optimization: randomization, gap estimation, and numerical resolution,

J. F. Bonnans, K. Liu, N. Oudjane, L. Pfeiffer, and C. Wan, “Large- scale nonconvex optimization: randomization, gap estimation, and numerical resolution,”SIAM Journal on Optimization, vol. 33, no. 4, pp. 3083–3113, 2023

2023
[19]

Convex optimization: Algorithms and complexity,

S. Bubeck, “Convex optimization: Algorithms and complexity,”Foun- dations and Trends in Machine Learning, vol. 8, no. 3–4, pp. 231–358, 2015

2015
[20]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar ´e,Gradient flows: in metric spaces and in the space of probability measures. Springer, 2005

2005
[21]

R. B. Vinter and R. Vinter,Optimal control, vol. 2. Springer, 2010

2010

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [13]

Deterministic and stochastic frank-wolfe recursion on probability spaces,

D. Yu, S. G. Henderson, and R. Pasupathy, “Deterministic and stochastic frank-wolfe recursion on probability spaces,”Mathematics of Operations Research, 2025

2025

[2] [14]

Frank-wolfe recursions for the emergency response problem on measure spaces,

D. Yu, S. G. Henderson, and R. Pasupathy, “Frank-wolfe recursions for the emergency response problem on measure spaces,”arXiv preprint arXiv:2507.09808, 2025

arXiv 2025

[3] [1]

Mean-field models in swarm robotics: A survey,

K. Elamvazhuthi and S. Berman, “Mean-field models in swarm robotics: A survey,”Bioinspiration & Biomimetics, vol. 15, no. 1, p. 015001, 2019

2019

[4] [2]

Mean-field optimal control,

M. Fornasier and F. Solombrino, “Mean-field optimal control,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 20, no. 4, pp. 1123–1152, 2014

2014

[5] [3]

Continuum swarm tracking control: A geometric perspective in wasserstein space,

M. Emerick and B. Bamieh, “Continuum swarm tracking control: A geometric perspective in wasserstein space,” in2023 62nd IEEE Conference on Decision and Control (CDC), pp. 1367–1374, IEEE, 2023

2023

[6] [4]

Mean field deep reinforcement learning for fair and efficient uav control,

D. Chen, Q. Qi, Z. Zhuang, J. Wang, J. Liao, and Z. Han, “Mean field deep reinforcement learning for fair and efficient uav control,”IEEE Internet of Things Journal, vol. 8, no. 2, pp. 813–828, 2020

2020

[7] [5]

A machine learning framework for solving high-dimensional mean field game and mean field control problems,

L. Ruthotto, S. J. Osher, W. Li, L. Nurbekyan, and S. W. Fung, “A machine learning framework for solving high-dimensional mean field game and mean field control problems,”Proceedings of the National Academy of Sciences, vol. 117, no. 17, pp. 9183–9193, 2020

2020

[8] [6]

Mean-field control barrier functions: A framework for real-time swarm control,

S. W. Fung and L. Nurbekyan, “Mean-field control barrier functions: A framework for real-time swarm control,” in2025 American Control Conference (ACC), pp. 1–7, IEEE, 2025

2025