High-dimensional Mean-Field Games by Particle-based Flow Matching
Authors: Jiajia Yu
Junghwan Lee
Yao Xie
Xiuyuan Cheng
Venue: ICLR 2026 Poster (2026)
Keywords: mean-field games
particle-based
flow matching
simulation-free
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: A particle-based simulation-free deep flow matching method for solving high-dimensional Mean-Field Games, with proved sublinear convergence.

Abstract: Mean-field games (MFGs) study the Nash equilibrium of systems with a continuum of interacting agents, which can be formulated as the fixed-point of optimal control problems. 
They provide a unified framework for a variety of problems, including both potential and non-potential games, with applications in areas such as generative modeling.
Despite their broad applicability, solving high-dimensional MFGs remains a significant challenge due to fundamental computational and analytical obstacles. 
In this work, we propose a particle-based deep Flow Matching (FM) method to tackle high-dimensional MFG computation.
In each iteration of our proximal fixed-point scheme, particles are updated using first-order information, and a flow neural network is trained to match the velocity of the sample trajectories. 
Theoretically, in the optimal control setting, we prove that our scheme converges to a stationary point sublinearly, and upgrade to linear (exponential) convergence under additional convexity assumptions.
Our proof uses FM to induce an Eulerian coordinate (density-based) from a Lagrangian one (particle-based), and this also leads to certain equivalence results between the two formulations for MFGs when the Eulerian solution is sufficiently regular.
Our method demonstrates promising experimental performance on MFGs in high dimensions.

Published as a conference paper at ICLR 2026
HIGH-DIMENSIONAL MEAN-FIELD GAMES BY PARTICLE-BASED FLOW-MATCHING
Jiajia Yu', Junghwan Lee", Yao Xie Xiuyuan Cheng' 1 Duke University," 2 Georgia Institute of Technology.
ABSTRACT
Mean-field games (MFGs) study the Nash equílibrium of systems with a continuum of interacting agents, which can be formulated the fixed-point of optimal control problems. They provide a unified framework for a variety of problems, including both potential and non- non-potential games, with applications in areas such as genera- tive modeling. Despite their broad applicability, solving high-dimensional MFGs remains a significant challenge due to fundamental computational and analytical obstacles. In this work, we propose a particle-based deep Flow Matching (FM) method to tackle high-dimensional MFG computation. In each iteration of our proximal fixed-point scheme, particles are updated using first-order information, and a flow neural network is trained to match the velocity of the sample trajectories. Theoretically, in the optimal control setting, we prove that scheme converges to a stationary point sublinearly, and upgrade to linear (exponential) convergence under additional convexity assumptions. Our proof uses FM to induce Fulerian coordinate (density- based) from a Lagrangian one (particle-based) and this also leads certain equivalence results between the two formulations for MFGs when the Eulerian solution is sufficienlly regular. Our method demonstrates promising experimental performance on MFGs in high dimensions.
1 INTRODUCTION
Mean-field games (MFGs) (Lasry & Lions] 2007 Huang et al. 2006) study the Nash Equilibria in games involving a continuum of indistinguishable non-cooperative players. In an MFG, individual cost is affected by the aggregate behavior of the population due to interaction effects. Given the collective behavior of the population, every player seeks an optimal strategy é that minimizes their individual cost. However, as players update their strategies to é, the overall population distribution also evolves to induced by é. mean-field Nash equilibrium (MFNE) this context is a state where the strategy chosen by each player is optimal with respect to the population, and the population itself is consistent with these strategies.
Mathematically, such an equilibrium can be formulated as the fixed point of optimal control problem Argmin(a.))CC,, J (p, ū; p), where the cost J as a functional the players' strategy and individual distribution á (induced by ū) also involves an external population distribution p. The equilibrium is then defined as
(0,0) E Argmin J(p ū; p), p p= P,
(1)
(D.8)EC(..))
that is, when the optimal control solution ý and the population p coincide. We refer to (p, 0) as the best response to p. The constraint set C(p.r) essentially poses a continuity equation (CE), that is, satisfies the CE associated with ö. The specifics of C(p.v) and J will be given in the subsequent sections. In this paper, focus MFGs with deterministic dynamics, i.c., the agents' evolution determined solely by their initial positions and controls, without any stochastic noise. These are also referred as first-order MFGs, as the PDE defining the constraint set C(o.v) is of first order.
Thanks to the mean-field approximation, MFGs provide a more tractable framework for analyzing systems of large number of interacting agents. Since their introduction, MFGs have found appli- cations across various domains, including economics (Angiuli et al. 2023) Carmonal (202T). social sciences (Lee al. 2021), and engineering (Djehiche et al. 2017 More recently, MFGs have

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attracted growing attention due to their emerging connections with topies in machine learning, such as optimal transport
(Benamou & Brenier 2000), normalizing flows (Huang et al. 2023: Zhang &
Katsoulakis 2023), deep neural network training (E 2017). and reinforcement learning (Mondal et al. 2022] Angiuli et al. 2023). Motivated by applications, there is a growing interest in tackling the computation of MFGs, especially in high-dimensional space.
Solving MFGs encompasses and generalizes several key problems. In the cost T(p 0; p), when the terms involving P are the first variation of some potential functionals, the MFG problem (I) admits a variational formulation (Proposition 2.1). This class of problems is commonly referred to as potential MFGs or mean-field control (MFC) in the literature. On the other hand, when the cost (p. 0; )) is independent of the individual has no interactions with the population and the MFG problem T reduces to an individual optimal control (OC) problem. Notably, the trajectory- regularized normalizing flows for generative models (Huang et al. 2023 Xu et al. 2025) can also be formulated as an MFG. However, despite substantial recent efforts, solving high-dimensional MFGs remains challenging. Among others, one difficulty is due to their inherent fixed-point structure. In
game theory, it is known that naïve fixed- exxetppoitt iteration ( p(*++11 ((4) may not converge. A widely
used alternative is fictitious play (Cardaliaguet & Hadikhanloo 2017
p := (1 ag) )p()) +
ax,p(k), which converges for suitable aj E (0, 1] under certain assumptions (Yu et al. 2024) and has close connections to the generalized Frank-Wolfe algorithm (Lavigne & Pfeitfer (023). Despite
its theoretical appeal, directly implementing fictitious play in high-dimensional settings becomes impractical because computing the exact best response expensive. Nevertheless, it suggests that even when the objective changes at each step, moving toward the best response of the current step eventually leads to an MENE.
In this work, we propose an iterative neural method for high-dimensional MFGs that, at each step, jointly updates particle trajectories in Lagrangian coordinates and trains a Flow-Matching (FM) network to parameterize the velocity field. Specifically, motivated by fictitious play. Our approach limits the search to a local neighborhood around the current state instead of sceking global best response and averaging at each iteration done in classical fictitious play. To implement this scheme, we introduce a concurrent particle and FM update. In first update a set of particles directly using the gradient-based information. Next, we train a neural flow model up to match the velocity of the updated particle trajectories via a mean- squared loss as in FM (Albergo & Vanden-Eijnden 2023 Lipman et al., 2023] Liu et al.) (023). While our setting differs from the original setup of FM, where the endpoint distributions are given (accessed via finite samples) and the goal is interpolate to the target distribution from a source distribution, the FM step in our method has the effect of disentangling the sampled particle trajectories and ensuring that the marginal sample distributions are preserved (see Figure [ Theoretically, we prove the convergence rate of the proposed algorithm, and the property of FM to preserve marginal distribution is used in our analysis. In practice, the FM step allows vo to be trained over batches of sampled trajectories and accelerates convergence.
propose proximal best response scheme
each iteration,
We summarize our contribution as follows:
We propose a proximal fixed-point scheme that consists of a particle optimization step and a Flow Matching step concurrently in each iteration to solve high-dimensional MFGs. In particular, our scheme can solve potential MFG (including the optimal control setting) as well as general MFG by leveraging the fictitious play approach.
2. Theoretically, we prove that the particle-based proximal fixed-point scheme converges sublinearly and linearly with additional convexity assumptions. Our convergence rates hold in the optimal control setting, while some intermediate results apply to general MFGs. In particular, our proof uses "trajectory disentangling" property of the flow matching, which also allows us to obtain certain equivalence between the Lagrangian (particle-based and Eulerian (density-based) formulations, assuming that the Eulerian solution is sufficiently regular.
3. We apply the proposed algorithm to simulated and image datasets, including a non-potential MFG and an image-to-image translation task derived from a relaxed transport formulation within the MFG framework.
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cranking specy
(a)
(b)
Figure 1: Illustration of the trajectory disentangle effect by Flow Matching (FM). (a) and (b) show the sample trajectories the beginning of training and after 20 outer loops of training, respectively, where "optimized particles" stands for trajectories after particle update, and "velocity field" shows trajectories resampled from a learned velocity field by FM, see Algorithm T Theoretically, we prove that FM disentangles the trajectories and reduces the dynamic cost, while leaving the interaction and terminal costs unchanged.
Notations The notations are standard and detailed in Appendix A and Table 2 In particular, (" [0, 1J) denotes the set of time-dependent vector fields ç R x [0, 1] - Id with fo [|e:l]] <
00, where the inner-product (E, uojo0)) == So (Ee, mt) and induces the norm Ilxeoj0.1]
1.1 RELATED WORKS
Fictitious play solving MFGs Fictitious play, originally introduced in (Brown 1949 1951
is classical algorithm in game theory. Building upon this idea, (Cardaliaguet & Hadikhanloo) 2017) applied fictitious play to MFGs using PDE analysis, proving convergence the MFC setting, where the problem admits a variational formulation. The connection between fictitious play and the generalized Frank-Wolfe algorithm was later highlighted in (Lavigne & Pfciffer 2023). which leveraged this perspective to accelerate the algorithm and establish faster convergence rates in the MFC setting. This line of work was further extended to general voon-variational) MFGs in (Yu et al,, 2024). achieving similar acceleration without relying a variational structure. From computational standpoint, the algorithms in (Lavigne & Pfeiffer 2023 Yu et al. (024) are implemented via the PDE formulation of the MFG, specifically using the optimality conditions of the optimization problem in (I. This leads to a coupled system of Hamilton- -acioonn--a------a Jacobi-Bellmaa (HJB) and Fokker-Plaan (FP) equations. These methods rely on mesh-based solvers. which can provide accurate numerical solutions in low-dimensional settings but struggle to scale efficiently in high- -iee-niimnnional problems due to the curse of dimensionality.
Beyond PDE-based approaches, another line of work has investigated the connection between mean- field games (MFGs) and reinforcement learning (RL), using RL. techniques to implement fictitious play (Perrin et 2020 Guo et al. 2023 Magnino et al. (025). Most works in this direction assume a finite or discrete state space, such as a graph, which limits their scalability to continuous domains like Rd. Although discretization via meshing is possible, it suffers from the curse of dimensionality and becomes computationally infeasible in high dimensions. Moreover, many of these studies focus on infinite-horizz settings, where the problem reduces finding time-invariant equilibrium. In contrast, we consider a finite-horizon setting, which requires solving a time-dependent dynamic problem and poses greater computational and theoretical challenges. Specifically, focus finite time horizon rescaled to the interval [0, 1].
Flow-based methods and simulation-free methods To avoid discretizing the state space, recent works use neural networks to represent the trajectories of particles and define the objective directly in terms of network parameters. For example, (Ruthotto et al. 2020; Zhou al. 2025a Assouli et al. 2025) parameterize the value function with neural network; the particle trajectories are then induced by the value function via ODEs. (Huang et al. 2023) uses discrete normalizing flows to model particle transitions between time steps. A common limitation of these methods is that they assume variational formulation of MFGs, which only exists in special cases (e.g., potential MFGs or MFC). Even in those cases, the objective depends nonlinearly the population distribution p, which is coupled with the velocity field v through a PDE constraint. As result, training requires backpropagating through PDE ODE solvers, which is computationally expensive.
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Simulation- frree methods avoid solving PDEs ODEs during the forward process of neural network, and therefore simplify backpropagation and lead to significantly lower computational cost. These methods have recently been developed for stochastic optimal control (SOC), where the cost functions F and G are independent of the population distribution and the dynamics include stochastic noise. (Hua et al. 2025) derive analytical gradient expressions that avoid simulation and do not decouple the distribution from the control. Other works (Domingo-Enrie) et al. 2024 2025) use a decoupling approach, training the flow through control matching or adjoint equations. In comparison, our method addresses the first-order MFGs, which is a broad class of problems, and in particular, allows F. G to depend on p. Conceptually, our approach leverages the Lagrangian coordinates formulation of MFGs and trains the flow velocity field by directly matching to sample velocities.
Deep network methods for MFGs Beyond the works mentioned above, several other deep learning approaches have been developed for solving MFGs. (Lin et al. 202T, Gomes et al., 2023) formulate the problem as a minimax optimization and alternate updates between two variables to find saddle point. (Chen et al. 2023a Zhou et al. 2025b Fouque et al. 2025) adapt actor-critic methods from reinforcement learning to the MFG setting. (Chen et al. 2023b) uses the PDE formulation of MFGs and designs two neural network modules to force the HIB and FP together. Our approach adopts flow matching techniques in modern deep generative models, and we demonstrate the efficiency of our method on high-dimensional image data.
2 PRELIMINARIES
2.1 EULERIAN AND LAGRANGIAN COORDINATES
Let [o, 1] be the time interval and E P5 (IR") be a fixed initial distribution that has density, where P5 (Rd)- = (2 n ACRd). In MFGs, the distribution evolution ý induced by a control flow via the continuity equation. We restrict our focus to distribution-control pairs (p, in the constraint set
C(p.o) (0, 0,0 + V (po) 0, Ø(:,0) μ, ő E Cp. én E (pe)
(2)
Here Cp AC2 1; (P,( W2)) the set of absolutely continuous curves in Wasserstein space (Definition B. in the appendix). According to (Ambrosio et al., 2008 Thm. 8.3.1), for any p E Cp, there exists a Borel vector field v such that e E (p:) and the pair (Pt, vx) satisfies the continuity equation in the sense of distributions. From a fluid dynamics perspective, v(x, f) describes the motion of fluid by observing the change at fixed spatial location over time and is known as the Eulerian coordinate. It is analogous monitoring the population density a fixed state æ over time. Due to its reliance on fixed spatial grid or mesh Eulerian simulation becomes impractical in high dimensions.
Alternatively, one may adopt a particle-based perspective by considering X E Cx (0, 1; (w)). Here, (xx.)) [0, 1] + Rdd represents the trajectory of an individual agent starting from position x with X(x, 0) x. For any a.e. I, the weak derivative (.,4) exists and is in (1). When all particles evolve under the same velocity field, the Eulerian and Lagrangian descriptions are linked by the ordinary differential equation:
0,X t) = v(X(z,t).t
(3)
and X is known the Lagrangian coordinates. Let C(x,,) be the constraint set where the ODE is satisfied for μ-a.e. x and a.e. t:
(X, i) E C(X,,) = ={ (xx,6) (x, = (x(( t), X(x =2 xx
(4)
Notice that for X E Cx, the trajectories may intersect, and there may not exist v such that (X,v) v) C(x.))- In Section 4 we show that by flow matching, there exist (p, v) E C(a.v) such that v takes average on intersecting points and Pt = ( On the other hand, given (p, v) E C(p.n). the ODE solution X to may not always exist and may not be unique if it exists. Let C. be the set of functions v Rd x [0. []] Rd that are bounded and Lipschitz continuous in & on every compact set B C Rd. Then, by standard existence and uniqueness theorem for ODEs (see, for example, (Ambrosio et al. 2008 Lemma 1.4),, there exists unique solution X such that (X, v) E C(x.v) and (X;) = Pt. In the literature, the Lagrangian formulation has also been considered in a weaker sense using probability measures on path space. We briefly discuss this perspective in Appendix B
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2.2 MEAN-FIELD GAMES
Given a population distribution evolution p E Cp. the individual cost of adopting strategy (ő. 8) depends on p. Precisely, Fe, G P2((R)) F - R be operators acting the population measure. For Pt E P2( F F(p]] R² - IR is a function on I It assigns an interaction cost 4 [p:](2)) to the individual at position x: at time given the population distribution Pt. The total individual cost in response to p is defined by
0)) == S.
e)1 +loxl(a)) (p)())dt+ Gloi)e)dā
(5) A pair (p. 8) that minimizes [(p, 0; p) over (p. E C(a.o) is called the best response to p. The objective in an MFG is to solve ( to find an MFNE, which is a triple (ø. 0; p) such that the individual best response is consistent with the population, i.e., X p. In the objective (5), we refer to F as the interaction coupling and G as the terminal coupling, where Flpt] and Gipi] represent the interaction cost and terminal cost, respectively. The couplings F,, G are typically either local, of the form (z) = f(((z)) with f R25 + R and p being the density of p. or nonlocal of form (z) = (K₂ (z) with K₂ being the kernel at
When the couplings F, G are L² first variations of functionals F,G on P2 (RR)) (5) takes the form as
J(p 8; p) = J.' / (1 t)1? D=IP)(()) dir(z)dtt + / D()
(6)
and the formal definition of the derivatives D.F, DG can be found the appendix (Definition B.2 In this case, the MFG problem ( is associated with a variational form (Lasry & Lions| 2007 Acchdou et al. 202T). We summarize the result the following proposition, and include proof in Appendix D. for completeness. Note that our definition of convex functionals on P(R4) is with respect to affine interpolation (Definition B.3)
Proposition 2.1. Let F P2(RR)) IR U {+00} and 9 P2(R2) Rd U {+00} be proper and consider the following optimization problem:
T(p.v) : [ 210o,,, 1o(0 (1(0,,0) dp) [ F(pi)dt +(p.)dt+((pp))
inf (p.))CC(oo))
If FF G have first variations F, G (2(RR) Rd - } under L22 metric, and (p. 6) Cla.v) is the minimizer to 7 then (ò. 0) solves the fixed-point problem ( If. in addition, F, 9 are convex in p. then (p, 0) is the minimizer to 7 if and only if (p: 6) solves the fixed-point problem 1
The variational problem 77 is referred to as a potential MFG or mean-field control (MFC) problem. F and G are referred to as interaction (potential) cost and terminal (potential) cost. The variational structure facilitates the usage of many optimization algorithms and connects the MFG (T) several important optimization problems. For example, when the interaction cost is zero, and the terminal cost a Kullback-Leibler (KL) divergence, the MFC
(7) reduces to normalizing flow regularized by transport cost (Onken et al. 2021 Huang al. 2023 Furthermore, since the KL divergence is convex, solving the MFG (I) is sufficient to solve the regularized normalizing flow. In the case of zero interaction cost and general terminal costs G, the MFC 7 recovers the
(JKO) scheme (Jordan et al., 1998) for Wasserstein gradient flows, which has applications in both generative modeling (Xu et al. (023) and solving high-dimensional kinetic equations (Huang & Wang 2024). When F' and G are functions on Id and independent of p. the game is potential with F()) (x))d((x) and ((p) I G(x) G(z)dp(x) linear in p. and both (T) and (7) reduce a single player optimal control (OC) problem. The costs F and G being linear in makes it convenient to approximate them with expectations.
2.3 FICTITIOUS PLAY
Fictitious play is classical algorithm in game theory, first introduced in (Brown 1949 later adapted to MFGs in (Cardaliaguet & Hadikhanloo 2017). The update rule is given by:
1951 and
(Qlk), (((k) E Argmin [ 8: (b)),
(p.))CC())))
(8)
(((+1) = = =(1-au)p + (k)
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When ag 1, the update becomes fixed-poin iteration, which is shown to potentially diverge Yuu
et al. 2024 Sec. 5.1). For potential MFGs, (Cardaliaguet & Hadikhanloo 2017 proved the
convergence using diminishing weight ag &. Further, (Lavigne & Pfeiffer 2023) showed that
fictitious play is equivalent to the generalized Frank-Wolfe algorithm for potential MFGs, and proved
convergence rate of P)) for 0g I+pp (with p 0). More recently, ( Yu et al. 2024) extended
the same convergence rate general (non-potential) MFGs and highlighted that the best choice of
the weight ag depends on the local convexity of the dynamic cost.
3 METHOD
We first reformulate the MFG problem (1) in Lagrangian coordinates and express the objective as an expectation. We then propose proximal fixed-point algorithm that alternates between updating the particles X and the velocity field v. Our method utilizes optimization over sample particle trajectories, namely particle-basede which is more scalable high-dimensional spaces.
3.1 REFORMULATION LAGRANGIAN COORDINATES
Consider characteristic maps X F' [0,1] R in the set Cx. For X Cx, X(x.)) denotes the trajectory of sample particle starting from x. For given p E Cp. define the individual control objective in MFG
J(X: == : ((10010)(000 (1,
(C de+ G[ii((((x))))
(9)
and population objective in potential MFG
[ (10xx(0))))) S. F((X))m)ddt (((8))
J(X) = E= Beoe
(10)
Using the Lagrangian coordinates, the MFG ( can be formulated as
X
Argmin J(X:P), Pt = À. = (Xt) #11.
(11)
XeCx
Similarly, the potential MFG
7 becomes
min J(X).
(12)
XeCx
Notice that in the reformulation, we only require X E Cx, that is, each trajectory itself is differentiable curve, but among each other the trajectories may intersect. In other words, for general X Cx. there may not exist a flow i such that (X, 6) C(x.)). In section 4 we will resolve this issue using Flow Matching (FM): FM will provide velocity field ō such that the marginal density of X satisfies the continuity equation associated with 6. In addition, this pair (ă, 8) will preserve or lower the objective I compared to the objective on X (Lemma 4.1 This result will further allow us to establish certain equivalence between the Eulerian and Lagrangian problems under regularity conditions.
3.2 PARTICLE OPTIMIZATION
A direct adaptation of fictitious play to the Lagrangian formulation suggests computing the best response (k) (kk) and then update (k+1) (1 0x) x x
(k) and ((-1))
) #1. However, solving for the best response can be computationally expensive. Moreover, when the step size a4 is small, accurate best response is often unnecessary, as the effect of any approximation error is scaled down by ag.
(x (x-(+++))
By definition, the best response provides a descent direction for the current objective J(:: p (*)). A key observation from fictitious play is that convergence to fixed point is possible long as each update yields a small improvement relative to the current objective, even though the objective itself changes at each iteration. Motivated by this, search for the proximal best response each

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iteration and propose a proximal fixed-poiie scheme. Specifically, given (1) and step size ag 0, the approach consists of the following steps (we write in the form of integration over distributions here, and in practice implement via finite samples and time discretization):
Trajectory computation: for initial point x which observes the distribution μe, solve for its trajectory driven by g((k) obtain ( such that (X(*), v(() C(x,o)-
2. Cost and particle update: update the population-dependeen cost functional :(x:p(()
(k) where Pi is the law of X (x; 1) with & 14, and then update X by proximal descent
X ( argmin ((X:p(o) + 20r
(IX- ( ( 11 + 1|x1 - x1 12) (13)
X
3. Flow matching update: update v by regression
(k+1)) = argmin Ezwp [//"
(z, (2,
0,x
3) (xx,
dt
(14)
In Lemmas and 4.5 we prove that both the particle update (13) and the flow matching update (14) are well-defined. When (*)), is not a stationary point, the particle update yields a decrease in the objective function. More importantly, if ( is not induced by a velocity field v, then the subsequent flow matching and resampling steps also result in decay of the objective. Based on this observation, we perform the flow matching update less frequently in practice, as it contributes improvement primarily when particle trajectories intersect.
In practice, the proximal descent update of particles (13) is implemented by several iterations of gradient descent. To derive the descent direction, consider smooth perturbation Y I2d [0, T + Rd satisfying Y(x, 0) O. For sufficiently small c, first-order expansion of J(X + pY;p) at
e yields p) + (Io' (Y,, - (Oxx x), + V Flpp:((x.). dt + (Yi, (0,X), + VG
The coefficient characterizes the descent direction. In particular, at intermediate times f E (0, 1), the particles evolve along the direction (Qux)): : [F][(:)(),) while at the terminal time, they follow
- -(O,,X) x V[[p1](X1)) Consequently, each cost update requires evaluating the functionals F(p:] and G(p1). together with their (sub) gubb)gradients. The specific form of the update depends on the choice of F and G. We provide two representative examples in Examples 3.1 and 3.2
de Example 3.1. When G(p) KL(p)e [p), we have G[p) log Je. In our particle-based method, are to update the cost Glp) when the distribution p (and possibly also v) is given by finite samples. An estimator of G(p) which provides its (sub)gradient any query point
fulfill the need. In our experiments, we use neural network classifier to approximate G[p).. and the gradient is computed via auto-diiferreniiatio..
Example 3.2. A nonlocal coupling is of the form (x)) = wwa y)dp(y) where w : Rd xRd + IR is a kernel function. w(z, y) is differentiable in æ, then (x) = f VV w(z ydd())) Both F(p) and V [p] can be approximated via empirical averages over particles.
For the flow matching update, we parametrize v by neural network uo and (14 is implemented using standard stochastic optimization schemes such as Adam. The complete procedure is summarized in Algorithm and more details are given in Appendix
4 THEORY
We begin by noting that the original problem is posed in the Eulerian formulation ①.. while our algorithm operates in the Lagrangian formulation (TT). In this section, we first derive a certain equivalence between the two formulations in Theorem[4.3 In Section(4.2]
analyze the convergence of the proposed scheme [4), proving its descent property under for general MFGs and its convergence rates in the optimal control setting.
4.1 EOUIVALENCE BETWEEN EULERIAN LAGRANGIAN PORMULATIONS
To prove Theorem 4.3
we first relate X E Cx to a corresponding pair (ū, 0) E C(p.v) and compare their associated costs. Lemma 4.1 shows that for any X Cx, there is unique pair (ā. 0) C(o.v)
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Algorithm 1: Particle-based Flow-Matching for Mean-Field Games
Initialize the velocity field Up
for k = =1,2, 1,,,..., K do
Trajectory sampling: sample a batch of the initial point (X,0) and compute trajectories
X [Xs,t, 0 و m] by intergrating the ODE using the velocity field to;
Cost and particle update: update X using cost functionals F and G by (23) for L1 steps;
every Lo steps, update F and G from the current trajectories XB; Flow matching: update vi to match the trajectories XB by minimizing the loss 24 for L2
steps:
end
obtained by flow matching. In addition, the cost of is bounded below by the cost of (Ø, 0). In the other direction, Lemma 4.2 establishes that the equality holds when X is induced by velocity
field 7 E C.. These two lemmas will also be used in the convergence analysis in Section 4.2 and
specifically, to prove Lemma 4.5 The proofs of Lemmas 4.1 and 4.2 are given in Appendices D.2 and D.3 respectively.
Lemma 4.1. Let X E Cx and set ū argmin, Ex-n [o (1 011) (2x 7. (H. Then, P e Cp: (p, C(p.)), and J (; S J(X: p) for any p Cp.
Lemma 4.2. (Ø, 6) E ((..v) and i E Co, then there exists X E Cx satisfying (X, 0) E Cx and for any p E Cp. ((::p) [p, p).
As direct result of Lemmas 4.1 and 4.2 the following theorem shows the relationship between Eu- lerian coordinates and the Lagrangian coordinates. Specifically, under certain regularity conditions, we show that for general MFGs, an Eulerian solution induces a Lagrangian solution; For potential MFGs, the optimal value of the Eulerian and Lagrangian formulation coincides, and a Lagrangian solution also induces an Fulerian solution by flow matching. The proof is in Appendix [..4
Theorem 4.3. (i) Suppose (p","") is a solution to the MFG T and v² E Co. then induces characteristics X Cx that is solution 11
(ii) Suppose the potential MFG 7 admits a solution (p",v*) with 1- E C.. Then v² induces characteristics X* E Cx that is solution to (2) and T(X*) = I(p", J In addition, for any other X E Cx that is also a solution to (1)) let (p. v) be the flow-matched Eulerian representation of X in Lemma 4.1 then (p. v) solution 77, and in this case J(X) [p. J"
4.2 CONVERGENCE RATE OF THE PROXIMAL FIXED-POINT SCHEME
In our scheme, the particle update 13 reduces the total cost; The flow-matching update (14) regularizes the trajectories reduce the dynamic cost while keeping the interaction and terminal costs unchanged. As result, the three steps together yield a decay in the objective value, as stated the following two lemmas. Proofs are provided in Appendix D.5 and D.6
2.53
by the proposed algorithm. Figure 2: Intermediate distributions between x 4 checkerboard and an isotropic Gaussian solved
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-0.1
9.0
0.3
0.2
Epech
t=0055
-0
10
2
-05
-0.
-0.5
-0.
(Tfp Residual defined in (57) versus the loop index k. (Top right) Test trajectories computed from the learned ug. Players move toward the target line 22 = -1, 1, adapting their speed to reduce the costs. (Bottom) Estimated density evolution (from test trajectories). The population tends to concentrate along 22 and spreads along I1 due asymmetric interaction.
Figure 3: Non-potential MFG in Section 5.2
Lemma 4.4. Assume that 1M
(1) G(ppi
are proper and L-smooth in R.d for any E [0, 1]. Then 13p admits a unique solution ( b) E Cx and
for any 0 < ak < t. the update scheme
[(x(++)) (()) & T(x(K) (())
(1xx(tt)) xx(k) 11 12oeto.11 + | (k+ x 1p²)
20k
(15)
Lemma 4.5. If ( E Cx, set Pt (x-1) = (x! (x Then, glk+1) E Cp and there exists (b+1)) solves (14) and 1t (k+1) is unique upto a pr-zero measure In addition, if u(k+1) Co. C(x,0), and for all such ((++1,,
then there exist (1++)
Cx such that (x(++1), wl(+1)) (x((t)),ubbt)) T(x(+t);; p) & T(x(+++);
For general MFGs, establishing how the descent property leads to convergence to fixed point is nontrivial and remains an open question for future investigation. In this paper, we show that the proposed scheme achieves sublinear convergence rate for optimal control problems (that is, where F, G are independent of p) and a linear convergence rate when F. G are additionally strongly convex on Rd The result is stated below, and the proof is in Appendix D.7
with O Theorem 4.6. Let the sequence ( ((k) generated by update scheme 13 14
< t, and assume (kk) E C.,, for all If F, G are independent of and are proper and L-smooth functions in Rd,, and J T(X)= 2 J for any X E Cx,, then X(k) satisfies
min (1x((+))) - (x 1iooee + (x+ xx*x KSK
S 2a(J(X(0)) - J)/K,
(16)
If in addition, F and G are X-convex (A > 0), and there exist solution (X+, p²) to then the sequence (k) generated by update scheme 13 and ((k) ( with 0 ag
satisfies
(x)
[1010111 + (|x((*) 111
Xi1 <
(I + 2 (IxI10) X* 1 + +X -xil
(17)
5 EXPERIMENTS
We consider two types of MFGs: Section 5. and 5.3 compute the transport between two distributions via an MFG formulation, and Section 5.2 considers a non-potential MFG. Additional experimental details provided in Appendix E and code can be found at wtthhp:::///g thub. www. nom/jiajja-yuj mfg flow wwwoooowaooooo matching
5.1 Toy EXAMPLE
We are to learn transport in R from 4 x checkerboard distribution to an isotropic Gaussian. In de this case, F[p] 0 and G[p] log dr as in Example 3.1 where is (N( I). Figure 2 visualizes the evolution of the distribution, where the learned transport map continuously interpolates between the source and target densities.

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Randomly selected four OT trajectories for image-to -ongge translation of handbags shoes (left) and CelebA male + CelebA female (right).
Figure 4:
Table 1: FID of translated images of our method and baselines. Lower FID is better. Results for
Disco GAN, Cycle GAN, and NOT are quoted from Korotin et al. (023), and results for Q-Flow from Xu et al. 2025).
Ours
Q-Flow*
OT-CFM
SI
Rectified Flow
SB-CFM
Disco GAN"
Cycle GAN*
NOT
Handhag
Shoes
12.44
12.34
13.01
15.87
13.91 14.01
12.70
22.42 35.64
16.00
13.77
Male
Female
9.68
9.66
12.88
16.39
11.55
17.74
13.23
5.2 NON-POTENTIAL. MFG
We consider an MFG on R² with interaction cost (1r] = Ap w(x y)dp(y), where w(x, exp(a T (x y)) and a E IR2 is a fixed non-zero vector. The is the smoothing coupling as in Example 3.2 Since w is asymmetric, the game is non-potential. The terminal cost is G(x) The terminal cost independent of p. and gradient has a closed-form evaluation. More details can be found in Appendix E.2. Figure 3 (top left) shows that the algorithm converges to an equilibrium state with a residual of 10 The learned flow visualized the top right plot on test trajectories. The bottom plot shows the density evolution. The results show that players move toward the terminal line a x c to reduce terminal cost, and as players evolve, the population compresses along the direction of and spreads along a making y nearly orthogonal to This density evolution aligns with the strategy matching population pace while progressing toward a x =
Àa(a c)² where с E IR, encouraging players to move toward the hyperplane a
5.3 IMAGE-TO-IMAGE TRANSLATION
We formulate the image-to-image translation task as a transport from source to target distributions under the MFG framework, by employing a KL divergence terminal cost, same the toy example.
We consider two tasks: (i) Handbags + Shoes Yu & Grauman 2014 Zhu et al. 2016), and (ii CelebA male + CelebA female (Liu et al. 2015). In each case, the objective to generate target-domain images conditioned on source-domain inputs. Following |Rombach et al. 2002)). we first train a deep variational autoencoder (VAE) to embed images into a latent space, and all methods Iearn transport dynamics in this latent space. We evaluate performance using the Fréchet Inception Distance (FID) (Heusel et al. 2017); implementation details and baselines are provided in Appendix E.3 As reported in Table our method achieves FID scores comparable to Q-Flow and outperforms other baselines. Figure shows representative translation trajectories, illustrating smooth transitions with improved color consistency and fewer visual artifacts.
6 DISCUSSION
The work can be extended in several future directions. First, the particle-based method incurs significant memory costs when the number of trajectories and the number of time steps are large. Second, while our framework applies to general MFGs without requiring a variational formulation, the theoretical convergence guarantees currently only cover the optimal control setting. Third, the convergence analysis considers the idealized proximal fixed point scheme, and it would be useful to develop complete analysis that accounts for practical sources of error, such sampling, finite difference approximations, and flow matching inaccuracies. Finally, it would be useful to extend the method to more datasets and real applications.
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ACKNOWLEDGMENTS
The authors thank Chen Xu and Edward Chen for their help with numerical experiments. The project was supported by the Simons Foundation (grant ID: MPS-MODL-000S partially supported by NSF DMS-223788 The work of JL and YX was partially funded by NSF DMS-2134037, 12M5-- and the Coca-Cola Foundation.
XC was also
ETHICS STATEMENT
We affirm that this research complies with the ICLR Code of Ethics. No conflicts of interest associated with this work.
REPRODUCIBILITY STATEMENT
The source code for the experiments publicly available. The authors clarified details all experiments for reproducibility the Numerical Experiments section or the Appendix.
THE USAGE OF LARGE LANGUAGE MODELS (LLMs)
The authors used a large language model (ChatGPT, based on GPT-4 architecture) assist with revising the writing of this paper. The model was used solely for improving the clarity, grammar, and fluency of the text, without contributing the conceptual, methodological, experimental content.
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A NOTATIONS
We summarize the notations in Table 2
A hat e denotes best response quantities. Let P2 (Rd) be the space of Borel probability measures with finite second moments. We use both the 2 and Wasserstein-2 (W2) metrics on (2((R specifying which is used when needed. For μ E P2 we define L²²(4) as the Hilbert space vector fields E Rd + Rd with / [[(x))(²d(((z) < co, inner product (E, n) = I ((x) ((())d((z) and the induced norm || I|ue The symbol denote a static distribution in (RR or time-dependent mapping [0, 1] + (IR)), with P: denoting the distribution at time t. Similar conventions apply v and X, which may refer to functions Fdd + 12 or time-dependent maps Rd [0, + Rd Constraint sets are denoted by C., where the subscript indicates the constrained variable. Objective functionals are denoted by J. For instance, J(p üt p) evaluates the cost of iistribution-velocii flow under fixed reference flow p. and T(X:p) p) evaluates the cost of characteristic map under fixed reference flow p.
Table 2: Summary of Notations
Notation
Description
P2(Rd)
Set of Borel probability measures on Rd with finite second moment. Space of functions E Rd - IRd such that I [[( du(x) < oc.
L²(H)
(E, 1)
Inner product on (1) defined as ((E(s), nxx)) du(x).
|
Norm on (n) defined as ((E, )
Lz(p @ [o, 1J)
Space of functions E Rdd x [0, 1] + id with finite μ @ Lebesgue-norm.
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Notation
Description
(5, p0.0011
Inner product on L2 @ [0, 1l) defined as So (E,, (Ei,m)) dt.
Norm (r [0, 1]) defined ((5/5))u(00.))
Fixed initial distribution in P2(R").
14
Population distribution. Depending on context, p P2(R or p:: [0, 1] +
P
Pz (RR),, ),RR)),, Pr = p(t).
Control function, velocity field. Depending context, R + RR
v: [0, 1] - (RR: Rd), 0z = v(().
x
Characteristic map. Depending context, X : Rd Rd or X : 00. 1] + L'(x), = (t).
m
Momentum, vector-valued Radon measure m = pv.
C
Space of absolutely continuous curves AC 1; (PP((1))) W2)) with finite quadratic energy.
Cx
Set of maps X Fd [0, 1] E2 such that X (x,,) e AC (0, 1; for j4-a.e. .r.
C
Set of functions u Id [0, F that are bounded and Lipschitz continuous in & on every compact set B C Rd
((a,))
Feasible set of pairs of distribution flow and velocity field (p. v). defined in (
C(x,))
Feasible set of pairs of characteristic map and velocity field (X, v). defined in 44.
C(.m
Feasible set of pairs distribution flow and vector-valued Radon measure flow (p, m) defined in (2
R(p,v)
Dynamic cost of (p, v) E ((.,))- R(p,v) = fo (dep Dynamic cost of X E Cx- R(X)= = So [10, d)))) Dynamic cost of (p, m) E C(p.m)- R(p,m) To dpc
R(X)
R(p, m)
dan (d)))
F
Interaction cost coupling, P(Rd) x Rd + R, (2, + Fr))(r). See
Exs.
3.1 3.2
Terminal cost coupling, similar to F. Individual optimal control objective on C(aw)) for given X E Cp, defined in 5
T(p,
J(X:P)
Individual optimal control objective on Cx for given ò E Cp. defined in Individual optimal control objective on O(p.m) for given P E Cp. defined in 27). Interaction and terminal potentials, P2(Rd) + R.
J(p, (p,m;:))
F,9
T),
MFC/potential MFG objective on Cía.v), defined in 7 MFC/potential MFG objective on Cx defined in (I0)
T(X)
mp
MFC/potential MFG objective on C(p.m) J(p = R(p.m) + J F(p.)d++ ((1)).
m, Dt,ts
m is number of time steps, At and fj = jAt. Approximate location of particle initialized at a, at time ty, i.c., Xt.1, ss
X1.4,
X X(,,,t) ty).
on
Weight for fictitious play in (8) or the proximal step-size in 13
B
Step-size for particle update in (23
K
Total number of outer loops in Algorithm
B
Number of samples per outer loop.
Lo
Frequency of cost update.
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Notation
Description
L1. L2
Number of particle update iterations and flow neural network update itera- tions, respectively.
B ADDITIONAL PRELIMINARIES
Absolutely continuous curves In our setup, we ask p to be an absolutely continuous curve in (P2(R), W2) and (x, absolutely continuous curve in IR with Euclidean distance for μla.e. x. We recall the concept of absolutely continuous curves as defined (Ambrosio et al. 2008 Def. 1.1.1).
Definition B.1 (Absolutely continuous curves). Let (X, d) be complete metric space and let
:: (0, 1) + X' be a curve. We say that 1 E (0; if there exists a function m E (2 1) such that
d(I(s),((t) & J m(r)dr, vO < 8 <<sst<
(18)
By Ambrosio et al. 2008 Thm. 1.1.2), for any / E AC X),,, the limit
d((()).((())
(1)( == lim 8-8
(19)
|s 1--th
exists for a.e. E 10. In addition, |1'] E (0, 1) and |t" is admissible integrand of (88) and for any m satisfying (18). (1) S m() for a.e. t E (0, 1). [P'| is called the metric derivative.
First variation in probability space In Proposition 2.1 we show that when F, G are first variations of some F,G, the MFG (T) is associated to an optimization problem 77). The first variation in probability space is also called the Lions derivative. The main difference between it and Fréchet derivative is that the perturbation is required to have zero measure.
Definition B.2 (First variation (Cardaliaguet & Hadikhanloo 2017)). Let F: :2(((R)) + R, we say that DF P2 (TR.) x Rdd R is the first variation of for any P.Pl E P2(R").
lim (F((1 s)p + s - F(D)) DF - p)(x) 800 s
(20)
we say that DF (2(R)) x c + R is the first-order subdifferential F for any p. pf P2((R)).
F((1 s)p -p') - DFHola)d(d--le -
(21)
Since DJF is defined to additive constant, we assume that
f (FFel(e)d(())== = O, Vp E P2(Rd).
(22)
Convexity of cost functionals on P2(Rd)
Definition B.3. We say F PP( + R is convex if
F((1 - 8)p² + 6 s (1- (1-8))(6))
for any 5 E [0, pt E (2(R)..
Comparison with path-space formulations In the studies of optimal transport and optimal con- trol, useful extension of the classical ODE flow is via measures the path space (Superposition Principle (Ambrosio) 2008 Ambrosio et al. 2008)). which, e.g., allows branching of trajectories that follow the velocity field we parametrize the velocity field v by a neural network, which in practice usually gives a regular v. Thus, focus when is regular, and specifically when the solution to MFG has C..
weaker sense Lisini (006); Cavagnari et al. (2022). In this paper,
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We also comment on the results of Theorem 4.3 in comparison to the equivalence results (between the Lagrangian and Eulerian formulations) derived under the path space. For control problems, such an equivalence has been previously studied in (Cavagnari et al., 2022) by considering distributions on path space, and their Lagrangian formulation in a weak sense. In contrast, our framework is motivated by neural network methodology, and focus our theory on classical ODE flows where trajectories from jea.e. x are well-defined, which is possible when assuming the MFG solution has a regular v².
C ALGORITHM DETAILS
We provide details for Algorithm to implement the proposed method in Section 3
The velocity field v is parameterized by neural network uo, and X is represented by time-discretized sampled trajectories (xx.)). Specifically, let At 1/m and define the discrete time grid by t, j&t for j ,,..., m. The trajectory of the i-th particle is denoted { where X1.,, X (2,, ty), and z3 the initial point drawn from distribution p.
Initial velocity field Before the outer loop, we need to specify an initial value for ug. When there is prior knowledge, we can use it to prescribe the initial velocity field. For example, v(z, () = 0 for all (z. t), when appropriate. We adopt this initialization for the non-potential MFG example in Section 5.2] Alternatively, one may first assign a target point to each initial sample and construct trajectories via linear interpolation between the initial and target points. The network vy can then be initialized through a flow-matching step on these interpolated trajectories. We use this strategy for the transport tasks in Sections 5.1 and 5.3 where the target points are sampled from the desired target distributions. This initialization provides favorable starting point, as it produces a terminal distribution that is already close to the desired target distribution.
Loop iterations
Below are the details for the loop iterations:
Trajectory sampling.
We first sample a batch B of initial points {r:}/= In our experiments, x; ~v is sampled on the fly if μ is analytically known; When only finite samples of μu are given (in a training set), we sample x, from the dataset. Given an ODE velocity field and an initial point 24, we numerically integrate the trajectory X (z,, t) using standard schemes, such fourth-order Runge-Kuute (RK4).
2. Cost update. The empirical distribution of the particle locations {Xi,,, Jizı approximates the density Pt, at time lj. and used to estimate the coupling functionals Flp,,] and Gipt...: Rdd + IR, along with their corresponding (sub)gradients. The resulting estimators depend the specific form of the coupling.
For Ex 3.1 when the classifier is updated frequently. the particle distribution changes gradually, so the classifier can be trained for fewer steps at each update.
For Ex 3.22 the functional F and its gradient admit representations terms of expectations, which can be approximated using empirical samples. In contrast, G and its gradient have closed-form expressions and can therefore be evaluated directly.
Particle update. In practice, we update particle trajectories X =Xii,, 00 S m} for L1 steps by:
X
j j= 0,
X1a, f
X1.,, BAt (-((uxx)),, + (VFI,,
j 1,2, ... 1, (23)
X1,tm 33 ((D,X,),,. + VG VG(pt...(X......)) (Xt,t.))
j == m.
Here +- indicates that X,,,, is updated by the right-hand side expression, ,3 is the step size, and Dr. and D, are finite-difference approximations of Òz and ô, respectively.
3. Flow matching update.
The flow matching update trains the velocity network vn to match the empirical velocities of particle trajectories B which is closely related flow matching methods (Albergo &
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Handen-Eijnded 2023
Lipman al. 2023
Liu et al. 2023). The training loss is
A
X1,t >
min
vo Xitt-1142 At (Xia;
(24)
We optimize 0 L2 steps using Adam. Since the particle update does not require access the neural network ve. and flow matching reduces the cost only when trajectories X intersect, we apply the flow matching update less frequently.
In each loop, we compute on the batch B of particles. The I,1 steps of partiele update of trajectories X can be computed by an innerloop mini-batches within B, and same with the Le steps of velocity field update by flow matching on X
Computational cost The main computational costs Aeporrtthmcome from the cost and particle update (line 4) and the flow matching update (line 5). The cost of the particle update scales as O(L))), and that the flow matching update scales as (2)),, both proportional to the batch size and number of update steps. Importantly, the flow matching update is performed infrequently. In addition, compared to simulation-bed based methods, our approach offers simpler and more efficient training process: the velocity network is updated via a decoupled, simulation-free flow matching step, which significantly reduces overall computational cost.
D PRooFs
D. PROOF oF PROPOSITION 2.1
The key for the proposition to hold is that, up to the change of variables (p.m) (p. pe), the continuity constraint becomes linear and the dynamic cost becomes convex. With this change of variable, it becomes straightforward that the fixed-point formulation ( is equivalent to the first-order optimality condition of the variational problem
Proof. Let w pe be the vector-valued Radon measure, and let dm denote its Radon-Nikodym derivative. Define
ap
Cpp.m) {(p, () 0,p V m O, Po n. C,C
(25)
The constraint is linear in (ū, m.). hence the set ((e.m) is convex.
Define the functional
1 ('') /
dif dp: (m)
2 2/11
dp,(s) my < P. VL [0,1],
R(x m)
(26)
OC,
otherwise.
Then R(Ø m)) is convex in (p, m). Next, set
J(p = m) ('' Flolld((o) / (lpi)ap(0).
(27)
The functional [pp mi p) is convex in m).
Since the objective is convex and the constraint is linear, (ō, m) solves
min (p.m)EC CCp
(28)
and only if it satisfies the first-order optimality condition:
0, +
2 = F(r), VV
L. = G(pi), =
(29)
0,++ V ग 0,
3o = μ.
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Here, the Hamilton-Jacobi-Beae Bellmmmn equation is understood to hold on the support of f. with equality replaced by inequality -0,0+ } Voll² < Flp). Φ1 G[pi] outside the support. The Fokker Fokker-Planck
equation is understood the sense distributions.
Therefore, (p, 6) solves if and only if it satisfies
- 0, + | F(J,, 31 Gl 1 VV.
(30)
0,6+ V (pi) 0. Po μ.
Similarly, deriving the first-order optimality condition for 7 yields the same system (30) Hence, (ø, 6) solving is necessary for solving 7 and becomes sufficient provided that F, 9 are convex in as in Definition B.3
D.2 PROOF OF LEMMA
We prove the Lemma in steps. We first show that the distribution flow ý induced by X is in Cp and the interaction and terminal cost are unchanged with the change from Lagrangian coordinate X to Eulerian coordinate p. As summarized in the following lemma.
Lemma D.1. Let X Cx and set P. (X,) Then E Cp. ( (F (x),( for any F I&d + Rd and F(p.) = F((X,) a p)) for any P2((R)) + RR
Proof. The costs are unchanged as direct result of the definition of pushforward.
We first prove that Po P2(RR). Fix t [0, 1]. For p-a.c. since X (2x is absolutely continuous,
we have
X t) = /' 0x,((,
(31)
Therefore
+2 0.X sd))
1 s 21 + f 110,8 s)] (32)
(x t)1 s 2||=|| +2
where the second inequality is by the Cauchy-Schwarz inequality. Since μ P2( and X (x, AC 1; for p-a.e. v, integrating over gives
/ [.))))pp(x)= = / [|*(( (1) $2 [0]1d)(()) + // |0, [x,x(x, < <00
(33)
therefore L. P2
For any 0 S < t S 1, consider the coupling (X,, Xx) between e. and p.. By definition, we have
[
W2
JRa 11 - (=12((c((()+=))
(34)
f [[()(z)) - {x(x))|aa(==== = 11, 0. a.x du(x))
Let
g(r) = 110, -1,x,(,,,))1,, 11a (11xxx(..)))))) 1
(35)
Then E (,0, 1)) and
We(ps, pi) < $/1 (.x((,))dr
1 = dr dr g(r)dr.
(36)
Therefore Cp-
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We then construct the velocity field 0 such that (0,0) ((..o) show that e is the unique solution to the FM, and the dynamic cost in (p, ö) is bounded above by the dynamic cost in X.
Proof of Lemma 4.1 We construct e and show that it solves the FM. Define the vector measure
(B == J (xX t)d((), B C BCR.
(37)
(x:8(x
Then my < Pt and by Radon-Nikodym theorem, there exist Borel vector field én defined pr-a.e. such that (B) = Sn ūz (y)) This implies that,
/ o(((z,)) 0,8 (dd(()dd 0,
(38)
Consider the FM problem
min J. [ [10X(((2,))
0,X - (d)())d)
(39)
38 shows that ē satisfies the optimality condition of 39 Since the problem (39) is unconstrained
with a convex objective, i is therefore the solution to
We then prove that (ø. ö) satisfies the continuity equation. By the chain rule,
dt / By the definition of i, for any Cg (TEd),
d ((y)dp(())> at / ((x(,, ()dp(x)= = / (x((((,)))
(40)
Vi(y)= . o(y t)dp:(y) = / (x((Xxz, 0,8(1, f)du(z).
(41)
Combining
and gives the desired weak formula the continuity equation: d dt f polōr / Vip vrdp.
(42)
In the end, we show
| S (,,
(43)
This implies ūg E (p:) and therefore concludes (p, 8) Cip.)- addition, integrating 43 over time and combining with Lemma D.1 gives (p, ū; p) S p():).
Define linear funational T' on (p:) by
T( (1)) dm. ((((2,))) t)) - (x,
(44)
Then by Cauchu-Schwarz inequality in (²²
[T(+) < || o xx t)l,, = 110,
(45)
which implies that T' is a bounded linear functional on the Hilbert space 2 (px) with operator norm
||T S 11,,8( (ll
(46)
By the Riesz representation theorem, there exists v, E (p.) such that for every up E L²())).
T(t) / (1()) (g))
(47)
Moreover,
| p. = |T < x
(48)
suffices to show that v, = ج.
Let c Rd be Borel set and a E Rd a fixed vector,
a, y B, By definition
then (.:( e (vv)= == (v
T(t) = mB( On the
0, $ B.
other hand, by Riesz representation (47). T(x) = a S (v)) Therefore a m ( =
JB (v)))((()) for any a E Rd and any Borel B c Rdd. This implies that e, dă, except on 7r-zero measure set, which concludes the proof.
dma, and v2 0
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D.3 PROOF or LEMMA 4.2
This is a direct corollary of (Ambrosio et al. 2008 Prop. 8.1.8).
D,4 PROOF OF THEOREM 4.3
Proof. We first prove (i). Since (p", 2*) solves T va is in C. and by Lemma 4.2 there exists
X E Cx such that (X²) ) # Pi and : = T(p" v²; v²:"") For any other X E Cx, by
Lemma 4.1 there exist (p, 8) E ((..)) such that [(p, p') S T(:: :"). Therefore, X solves
The same logic applies to MFC and concludes the first part of (ii).
For the second part of (ii,, it suffices to prove [ = [ vp. By optimality of (p. v), we
have T(p", =²) 2 vp, By Lemma (4.2), there is an X E Cx such that J(X) = v).
By Lemma 44. and optimality of X", we have T(p", S J(x) I T(X) [(, v), which concludes the proof.
D.5 PROOF OF LEMMA 4.4
Proof. The dynamic cost
1 _ 110,8(1, d = 110, 1o [0.
E Ex
(49)
2
is strictly convex in X, In addition, =lp! [ are L-smooth and as < t. therefore the objective of the update (13) is strongly convex. Therefore, the minimizer is unique it exists.
Let {X(0) Cx be minimizing sequence lim J J(X (x)); ((k) infxecx T(X; (k)).
Then by L-smoothness of Flp,")
( and G[pl)),
(,) and 0 < aa t. we have that
sup [10 (m) 1 [IX |1.ola + ||X ll.)) < 00;
which implies (x(")} is bounded in H (0, 1; (p)). Therefore there exists a subsequence, still denoted as X()), and X E H H 1; (n)) such that ( + X weakly in H H (0, 1; (r))..
By weakly convergence in H (0, (n)) and weak lower semicontinuity of I norm, ||a,x* 1| lim [|a (1)
By compact embedding theorem, there exists subsequence, still denoted as (() such that (() - X* strongly in C(O, 1; (T])) and therefore ( - X strongly in T,2 1; (1)) and x(") Xi strongly in (n))... Combining with the kinetic term, we have J(X;; (k)) < lim inf, n J(X(m);; (k)), which implies X* is the minimizer.
By the update rule, (k+ X and the decay property (15) holds.
D.6 PROOF OF LEMMA
4.5
Proof. By Lemma 4.1 p (k+1)
Cp. ii
(k+1) is unique upto pr-zero measure set.
Then by
Lemma 4.1 and Lemma 4.2 we have
T(x(++1); p) (p((t)) ,(ttt) s (x((+t)):
(50)
which concludes the proof.
D.7 PROoF OF THEoREM 4.6
Proof. Since F, G are independent of p. combining Lemmas 4.4 and 4.5 with as = a gives
2a ([x((+)))) - - 11 + (x+5))
x
(1) s ((x())) (x((++1 (51)
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Telescoping both sides and J(X) 2 J gives (16). If F, G are X-convex, then for any X, Y, E Cx,
(IX Yl + ||x1 - -Vill.)
$ 110:(x Y) + S.' (VF(X.) - VF(G), - -Y) + (VG(X)) - VI - YI)x
(52)
Take X ((++1) and Y X By the update rule and the optimality of X", we have
(1xx(++)- xx [00 + ||x((+)] -xil
(53)
< - 1 - (1x(6tt) - x(a), ((= -x") [000,1) + (X (x(+t)) - -x11) x! x((+1)
- -Xi))).
By algebra on the inner product
1 (1la 611 + ||a - c||² 116 cl12)
(a- b, a - c) = 2
(54)
Take a = x(++1), b = (1),
=== ,
X Then, (53 gives
(2da 1) ()(x(((+11) xx1 || | + ||x - xi
+ (Ix((tt)) 1
|| (+xx(t x l1t)
(55)
s (1xx2) x²1 || | + || xilli)
Therefore,
-x (Ixx
+ ||x - -xill.) S
1 + 2λo (Ix - -x + ||x -xill2)
-
(56)
and
17 holds.
E EXPERIMENT DETAILS
E.1 2D TOY EXAMPLE
We use multilayer perceptrons (MLP) with six hidden layers of width 512 and Rectified Linear Unit (ReLU) activations to model the velocity field and the classifier. The time t is concatenated with the input, and the concatenated input is fed into the velocity field.
E.2 NON-POTENTIAL MFG
In this example, we consider an MFG on IR2 with interaction cost defined by (z), t) AF I wwx y)dp(y) where AF = 10, w(x, y) = exp(a (x y)) and a = [o, 1] Since the kernel w is asymmetric, the game is therefore non-potential. Under this cost, player at state incurs exponentially greater interaction cost from masses located ahead of it in the direction of a than from masses behind. To reduce this cost, a player tends to move along the direction of a if many players are ahead of it. The terminal cost is chosen to be independent of the population density: G(x) (a T - c)2 where A 1, a is the same as that in F and c -1 is fixed. To minimize this terminal cost, players must move toward the target hyperplane a J = c. The initial distribution chosen as a Gaussian with mean [0, 1] T and a diagonal covariance matrix with entries 0.02 and 0.1. For this setting, evaluating the interaction and terminal costs and their gradients V F[p] G only requires computing empirical averages of e and does not involve neural network approximations.
To parametrize up. we use an MLP with 3 hidden layers of width [4, 8, 16). The hyperparameters: K = 100, B 1000, particle trained for In = 100 steps, and cost updated every Lo = 2 steps; flow matching trained for Le = 100 steps. Training performance is measured by the residual
Af
((axx ((x)))((x(3))
(8) Σ (D.x, )... + ( ||?
(57)
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which is sample- based approximation of the μ&(0. 1] norm of first-order variation of J [X;; p())) at X (x)),, The residual reflects how much the necessary condition of the MFG is violated; that is, a small residual indicates proximity to stationary point. Figure (top left) shows that the algorithm converges to a state with residual of 10-1.
We visualized the learned flow on test trajectories in the top right plot of Figure 3 and visualize the density evolution, estimated using Gaussian kernel density estimation in the bottom plot. All players move toward the line I2 = -1 to reduce the terminal cost. For players with initially larger 22 values, the interaction cost dominates early in the trajectory since there is more mass ahead of them along a. To mitigate this, they move quickly in the direction of a. Players starting at smaller 22 values also tend to move in the direction of a to avoid falling behind when other players catch up. However, because their initial interaction cost is lower, they balance interaction and dynamic costs, leading to slower initial movement. Although the initial distribution has a larger variance along the 22 direction, players' desire to reduce interaction cost causes them to compress along I2 and spread out along Ii after = 0.25, making x - y nearly orthogonal to a. Under this population profile, the interaction cost motivates agents to match the pace of the group while moving toward the target terminal location 22 = -1. The behavior is consistent with the physical interpretation of the costs and equilibrium.
E.3 IMAGE-TO-IMAGE TRANSLATION
We evaluate the quality of translated images by using Fréchet inception distance (FID) (Heusel et al., 2017) and compare with different baselines: flow matching methods including Optimal Transport Conditional Flow Matching (OT-CFM) (Tong et al. 2023), Stochastic Interpolants (SI) (Albergo
& Vanden-Fijnden 2023). Rectified Flow (Liu et al. 2023), Schrödinger Bridge Conditional Flow
Matching (SB-CFM) (Tong et al.) 2023); two representative generative adversarial networks (GANs),
Disco GAN (Kim et al. 2017) and Cycle GAN (Zhu et al.] 2017); Neural Optimal Transport
(NOT) Korotin et al. 2023); ; and Q-flow (Xu et al. 2025).
For image data, we first train a deep variational autoencoder (VAE) to compress images into a latent space. The deep VAE is trained the same setup following (Rombach et al. (022) in latent space of R 12x8888 enabling the trajectory to be learned in the latent space across all methods. We use constant learning rate of 4.5e-06 and a batch size of 32 with gradient accumulation over 8 batches for training VAE on both datasets. For the handbags-shoes dataset, a single VAE is trained for 100 epochs, using 90% of the images for training and the remaining 10% for testing. For the CelebA dataset, we train a single VAE for 100 epochs, adopting the standard training, validation, and test splits from forchvision datasets. CelebA. Table 3 reports the FID between reconstructed images from the training set and ground-trute truth images from the test set (handbags, shoes, CelebA male, and CelebA female), presenting the quality of the trained VAE.
We use convolutional U-Net architecture (Ronneberger et al. 2015) for the classifier. The Swish activation function used throughout the network, except for the final fully connected layer, where ReLU is used. For the velocity field, we use stack of convolutional layers with ReLU activations. The time is concatenated with the input along the channel dimension, and the concatenated input is fed into each convolutional layer. The same velocity field architecture is used for our method and the reproduced baselines reported. The detailed architectures of the classifier and velocity field are provided in Table and Table 3 respectively.
In our method, we initialize the velocity field using FM with a linear interpolant function. The hyperparameters: K 50, B =10k for handbag-tor shose and 15k for CelebA, particle trained for L2 = 1000 steps for handbag-to-shoe and 300 steps for CelebA, and cost updated every Lo = 10 steps; velocity field trained by flow matching for L2 1000 steps for handbag-to-she to-shoe and 300 steps for CelebA. For both the handbags-to-shoce. and CelebA datasets, the reproduced baselines (OT-CFM, Stochastic Interpolants, Rectified Flow, SB-CFM) are trained for 400K steps with a batch size of 512.
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Table 3: FID between reconstructed images from the training set and ground truth images from the held-out sets the images of Handhags, shoes, CelebA male, and CelebA female.
Handbag
Shoes
Male
Female
5.07
6.09
7.09
5.06
Table 4: Architecture details of the classifier.
Layer
Component
Conv 1
3 3 Conv2d, channels=256, kernel size=3, stride=1, hias=True padding=1
Group Normalization
num groups=4, channels=256
Conv 2
3 3 Conv2d, channels=512, kernel size=3, stride=1, bias= padding-1
Group Normalization 2
num groups=32, channels=512
Conv 3
3 3 Conv2d, 5hannels=512, kernel size=3, stride=2, bias= padding=1
Encoding
Group Normalization 3
num groups=32, channels=512
Conv 4
3 3 Conv2d, channels=1024, kernel size=3, stride=1, bias-True. padding=1
Group Normalization 4
num groups=32, channels=1024
Conv 5
3 3 Conv2d, channels=1024, kernel size=3, stride=1, bias=True, padding=1
Group Normalization 5
num groups=32, channels=1024
Conv Transpose
3 ConvTranspose2d, channels=1024, kernel size=3, stride=1, bias=True, padding=1
Group Normalization
num groups=32, 1024
Conv Transpose
3 ConvTranspose2d, channels=1024, kernel sizc=3, stride=1, bias=Truc, padding=1
Group Normalization
num groups=32, channels=1024
Conv Transpose
4 ConvTranspose2d, channels=512, kernel size=4, stride=2, bias=True, padding=1
Decoding
Group Normalization
num groups=32, channels=256
Conv Transpose
3 3 ConvTranspose2d, channels kernel size=3, stride=1, hias=True p
Group Normalization
num groups=32, channels=512
Cony Transpose
3 3 ConvTranspose2d, channels=256, kernel size=3, stride=1, bias=True, padding=1
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Table 5: Architecture details of the velocity field.
Layer
Component
Conv 1
3 3 Conv2d, channels=64, stride bias=True, padding=1
Conv 2
3 3 Conv2d, channels=256, stride 1, bias=True, padding=1
Encoding
Conv
3 Conv2d, channels=512, stride 2, bias-True, padding-1
Conv 4
3 3 Conv2d, channels=512, stride 1. bias-True, padding-1
Conv 5
3 3 Conv2d, channels=1024. stride 1, bias=True, padding=1
Conv Transpose
3 Con/Transpose2d, channels=1024, stride 1, bias=True, padding=1
Conv Transpose
3 x 3 ConvTranspose2d, channels=512, stride 1, bias=True, padding=l
Conv Transpose 3
4 x 4 ConvTranspose2d, channels=512, stride 2, bras=True, padding=1
Decoding
Conv Transpose
3 x 3 ConvTranspose2d, 2hannels=256, stride 1, bias-True padding-1
Cony Transpose
3 x 3 Con/Transpose2d, channels=64, stride 1, bias=True, padding=1
25


Reviews:
[{"review_id": "ztKTEIpdOf", "reviewer": "Reviewer_669D", "summary": "The paper proposes solving mean field games (MFGs) using a particle-based flow matching method. For a certain type of MFG, the proposed method replaces the backward best-response calculation and forward population simulation in a typical fixed-point iteration method with first-order gradient updates and particle-based flow matching. Theoretical convergence guarantees are provided for potential MFGs and the optimal control setting.", "questions": "- Why do you call the update rule (8) a \"trust-region strategy\"? Where is the trust-region radius, its update rule, and the rejection rule? Isn’t it a soft regularization method? Do you mean that you want to constrain the next iteration to be near the current iteration?\n- It is stated that the proposed method circumvents the \"main challenge in solving MFGs … enforcing the continuity equation constraint.\" Is it because it learns a velocity field that matches the trajectories, so it automatically satisfies the continuity equation? Am I correct in understanding that other works enforce the continuity equation by simulating a population consistent with the control, while you learn a control consistent with the population trajectories?\n- How are $n_{1}$ and $n_{2}$ related to $n$ in Algorithm 1?\n- What is the non-potential MFG considered in Section 4.2? Is it purely illustrative or a model for real applications?\n\nPlease also see the questions in the Weaknesses section.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper tackles the important challenge faced by fixed-point iteration methods for solving MFGs.\n- The proposed method is novel and reasonable.\n- The paper is mathematically rigorous and provides theoretical convergence analysis for the proposed method for certain classes of MFGs.", "weaknesses": "### Optimal Transport or Mean Field Game?\n\nThe model considered in this work is heavily inspired by optimal transport models but does not have much MFG flavor.\n\nFirst, the cost function has a very special structure: the running cost is separable into a control-related cost and a population-related cost. Furthermore, the control-related cost term is a simple quadratic function. These are two significant simplifications of the cost function that are not satisfied by general MFGs.\n\nSecond, a very special dynamics model, an Euclidean spatial state space with deterministic velocity control, is considered. This imposes strong assumptions, including continuity and determinism in the state evolution and that agents have full knowledge of the dynamics. General games do not satisfy these assumptions.\n\nThird, because of the simple dynamics model considered, agents' transitions are independent of the population distribution. This significantly reduces agents' coupling and thus weakens the model's connection to the general dynamic MFG framework, where both agents' rewards and transitions depend on the population distribution.\n\nThese model considerations and simplifications make sense from the perspective of optimal transport, but they substantially limit the applicability of the proposed method to general MFGs.\n\n### The Claim of \"Simulation-Free\" and Its Benefits\n\nIt is mentioned multiple times that the proposed method is \"simulation-free\", meaning that it does not simulate a PDE/ODE system. However, the trajectories of the particles still need to be simulated according to ODE (6). I do not see the fundamental difference between these two types of \"simulation\".\n\nFurthermore, it is claimed that other deep flow methods that simulate the population struggle in high-dimensional settings. However, it is not clear to me what the benefits of a particle-based flow matching method are in this regard. Specifically, the high-dimensional nature also requires an (exponentially) large number of particle trajectories $\\{ X_{i,t_{j}}^{(k)} \\}_{i,j}^{n,m}$ to approximate the true trajectory field $X^{(k)}$.\n\n### Related Work\n\nRecognizing the challenges faced by fixed-point iteration methods, several recent works \\[1,2,3\\] also propose simple methods that eliminate the forward-backward (best-response and population simulation) structure and demonstrate better convergence properties. These works are closely related to the discussions in the paper:\n\n- They are generalized Frank–Wolfe algorithms that update parameters using first-order gradient information. Thus, these methods do not need to calculate the exact best response or the induced population at each iteration. They share the same motivation that \"when the objective changes at each step, moving toward the best response of the current step eventually leads to an MFNE\" (Lines 78–79) and that \"convergence to a fixed point can still be achieved as long as each update yields improvement with respect to the current objective\" (Lines 293–294).\n- They are truly simulation-free methods, as the population distribution is learned in a model-free manner and can be estimated from a single trajectory of a single representative agent.\n- \\[3\\] addresses continuous state–action spaces by incorporating function approximation.\n\n\\[1\\]: Angiuli, A., Fouque, J.-P., Laurière, M., 2022. Unified reinforcement Q-learning for mean field game and control problems. Mathematics of Control, Signals, and Systems 34, 217–271.  \n\\[2\\]: Zeng, S., Bhatt, S., Koppel, A., Ganesh, S., 2025. Learning in herding mean field games: Single-loop algorithm with finite-time convergence analysis, in: International Conference on Artificial Intelligence and Statistics. PMLR.  \n\\[3\\]: Zhang, C., Chen, X., Di, X., 2025. Stochastic semi-gradient descent for learning mean field games with population-aware function approximation, in: International Conference on Learning Representations. PMLR.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ybbLDM88vq", "reviewer": "Reviewer_pZrr", "summary": "This paper proposes method for solving deterministic mean field games by combining a particle-based approach with a flow network by alternatively optimizing particle trajectories to minimize cost and the flow network to match the optimized particles. The proposed numerical scheme is shown to converge sublinearly for potential MFGs and linearly for optimal control problems under suitable assumptions. Numerical experiments on a non-potential MFG toy example confirm the convergence of the proposed scheme, while an image-to-image translation task on a learned latent space shows the proposed method produces smooth transitions and yields lower FID compared to baseline methods.", "questions": "- In relation to the above comment on the deterministic dynamics, it would be helpful to the readers for the author to cite and briefly discuss these additional works that do tackle this subclass, (e.g., [A, B]) and potentially compare against some of these methods (though not required), as well as ones that don’t make this assumption (e.g., [C, D, E, F]).\n- Minor: Typo with “following” in line 469\n\n[A] Gomes, Diogo, Julian Gutierrez, and Mathieu Lauriere. \"Machine learning architectures for price formation models.\" *Applied Mathematics & Optimization* 88.1 (2023): 23.\n\n[B] Assouli, Mouhcine, et al. \"Initialization-driven neural generation and training for high-dimensional optimal control and first-order mean field games.\" *arXiv preprint arXiv:2507.15126* (2025).\n\n[C] Lin, Alex Tong, et al. \"Alternating the population and control neural networks to solve high-dimensional stochastic mean-field games.\" *Proceedings of the National Academy of Sciences* 118.31 (2021)\n\n[D] Liu, Guan-Horng, et al. \"Deep generalized schrödinger bridge.\" *Advances in Neural Information Processing Systems* 35 (2022)\n[E] Chen, Yongxin. \"Density control of interacting agent systems.\" *IEEE Transactions on Automatic Control* 69.1 (2023)\n\n[F] Liu, Guan-Horng, et al. \"Generalized Schr\\\" odinger Bridge Matching.\" *arXiv preprint arXiv:2310.02233* (2023).", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "- To the best of my knowledge, the proposed method for solving deterministic mean field games is novel in the literature\n- The writing is clear and easy to understand\n- The empirical results help to demonstrate the effectiveness of the proposed method", "weaknesses": "- While the authors claim to solve (general) mean field games, it seems like the work only tackles **deterministic** mean field games, i.e., where the dynamics of each agent do not have any stochastic terms (e.g., equation in Line 170 doesn’t any diffusion related terms, and the evolution of the particles in (6) is an ODE and not a SDE). This could be somewhat misleading to readers, especially as this is not stated in either the title, abstract, or in the main body of the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JKl9DTTzU4", "reviewer": "Reviewer_HG67", "summary": "The paper is concerned with solving potentially high-dimensional Mean-Field Games (MFGs), which are challenging due to their optimal control fixed-point structure. Existing solvers often rely on mesh-based discretizations, and as a result become computationally infeasible when applied to high-dimensional problems. To overcome this, the authors propose a particle-based deep flow matching method along with a trust-region fixed-point scheme for efficiently solving high-dimensional MFGs.", "questions": "Given that there exist deep learning implementations for solving MFGs, would it be possible to compare against such methods (I was able to find some below)? I think this result would greatly strengthen the paper if indeed the proposed method converges orders of magnitude faster as hinted in the paper. \n\n\n[1] Chen, X., et al., *Physics-Informed Neural Operator for Coupled Forward-Backward Partial Differential Equations*. \n\n[2] Chen, X., et al., *A Hybrid Framework of Reinforcement Learning and Physics-Informed Deep Learning for Spatiotemporal Mean Field Games*.", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well written and easy to follow. \n2. The claims are justified with proofs whenever necessary.", "weaknesses": "1. Since the focus is on solving high-dimensional problems, out of the three example cases, it seems only the image-to-image translation results are noteworthy. \n \n2. Line 473: \"We can observe that our method produces smooth and coherent translations, particularly in terms of color consistency and reduction of visual artifacts.\" The authors only present the results obtained using their method, and this claim is rather relative than absolute. Hence, results using alternative methods are perhaps needed to support this claim/observation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)