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MeanFlow Meets Control: Scaling Sampled-Data Control for Swarms
Anqi Dong, Yongxin Chen, Karl H. Johansson, Johan Karlsson
Intelligence
Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 95%
Last extracted: 3/23/2026, 12:01:59 PM
Summary
The paper introduces a control-space learning framework for swarm steering, extending the MeanFlow approach to sampled-data control under linear time-invariant (LTI) dynamics. By learning a finite-horizon minimum-energy control coefficient rather than an instantaneous velocity field, the framework ensures consistency with the sampled-data structure of real control systems, providing a scalable solution for few-step swarm steering.
Entities (5)
Relation Signals (3)
Finite-Horizon Minimum-Energy Control → parameterizedby → Coefficient
confidence 98% · The learned object is the coefficient that parameterizes the finite-horizon minimum-energy control over each interval.
MeanFlow → adaptedfor → Swarm Control
confidence 95% · In this way, we place MeanFlow in a control framework and provide a scalable approach to few-step swarm steering.
Linear Time-Invariant Dynamics → governs → Swarm Control
confidence 94% · We propose a framework for sampled-data swarm control under linear time-invariant dynamics.
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Abstract
Abstract:Steering large-scale swarms in only a few control updates is challenging because real systems operate in sampled-data form: control inputs are updated intermittently and applied over finite intervals. In this regime, the natural object is not an instantaneous velocity field, but a finite-window control quantity that captures the system response over each sampling interval. Inspired by MeanFlow, we introduce a control-space learning framework for swarm steering under linear time-invariant dynamics. The learned object is the coefficient that parameterizes the finite-horizon minimum-energy control over each interval. We show that this coefficient admits both an integral representation and a local differential identity along bridge trajectories, which leads to a simple stop-gradient training objective. At implementation time, the learned coefficient is used directly in sampled-data updates, so the prescribed dynamics and actuation map are respected by construction. The resulting framework provides a scalable approach to few-step swarm steering that is consistent with the sampled-data structure of real control systems.
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- Source: https://arxiv.org/abs/2603.20189v1
- Canonical: https://arxiv.org/abs/2603.20189v1
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MeanFlow Meets Control: Scaling Sampled-Data Control for Swarms Anqi Dong Department of Decision and Control Systems, Department of Mathematics, and Digital Futures, KTH Royal Institute of Technology, Stockholm, Sweden. Email: anqid@kth.se. Yongxin Chen Institute for Robotics and Intelligent Machines, Georgia Institute of Technology, Atlanta, GA 30332, USA. Email: yongchen@gatech.edu. Karl H. Johansson Department of Decision and Control Systems and Digital Futures, KTH Royal Institute of Technology, Stockholm, Sweden. Email: kallej@kth.se. Johan Karlsson Department of Mathematics and Digital Futures, KTH Royal Institute of Technology, Stockholm, Sweden. Email: johan.karlsson@math.kth.se. Abstract Steering large-scale swarms in only a few control updates is challenging because real systems operate in sampled-data form: control inputs are updated intermittently and applied over finite intervals. In this regime, the natural object is not an instantaneous velocity field, but a finite-window control quantity that captures the system response over each sampling interval. Inspired by MeanFlow, we introduce a control-space learning framework for swarm steering under linear time-invariant dynamics. The learned object is the coefficient that parameterizes the finite-horizon minimum-energy control over each interval. We show that this coefficient admits both an integral representation and a local differential identity along bridge trajectories, which leads to a simple stop-gradient training objective. At implementation time, the learned coefficient is used directly in sampled-data updates, so the prescribed dynamics and actuation map are respected by construction. The resulting framework provides a scalable approach to few-step swarm steering that is consistent with the sampled-data structure of real control systems. Keywords: Flow matching, MeanFlow, sampled-data control, swarm control, minimum-energy steering 1 Introduction Large-scale swarm systems arise in robotics, autonomous transportation, distributed sensing, and collective exploration [BFB+13, CPD+18, DHF23, ZWW+22]. In such settings, the objective is to steer an entire population toward a desired configuration or distribution, rather than to plan individual trajectories agent by agent. For large swarms operating over wide regions, this viewpoint is naturally distributional, as it seeks control policies that drive an initial law to a target law in a dynamically consistent manner [CHE23, HKR21, RHC+23]. In practice, swarm control is implemented in sampled-data form [ACK12]: inputs are updated at discrete times and then applied over finite intervals. This distinction is negligible when the controller is updated at a high frequency, but becomes crucial in few-step control, where each control action must remain in effect over a larger time interval [BOU16, CYV+25]. Recent advances in flow-based generative modeling face a closely related issue. These methods transform a reference distribution into a target distribution through time-dependent dynamics over a prescribed horizon. The learned object is typically instantaneous, such as velocity fields in flow matching [LSK+25, LCB+23] or score fields in diffusion and score-based models [DVK22, EGP25, SSK+21b]. The training process samples an intermediate time and state, and regresses an instantaneous label induced by a training bridge [ABV25, LGL22], while execution takes place over finite windows. When many small steps are available, this mismatch remains limited, whereas under a few-step budget it becomes fundamental, since coarse discretization and repeated composition amplify local errors [CCL+23, LZB+22, SH22, SME21a]. MeanFlow [GDB+25a] addresses this mismatch by learning a window-level quantity rather than an infinitesimal one. Instead of modeling instantaneous velocity alone, it introduces an average velocity over each time window, namely the quantity actually used by coarse-time updates. This viewpoint has since been further studied from the perspectives of stability and robustness under coarse composition [GLW+25b, HLM+25]. The key idea is that when execution is window-based, the learned object should match the quantity applied over that window. In control theory, this principle may be adapted to structured dynamics. In many problems, admissible motion is not specified by an arbitrary vector field, but by a state equation with homogeneous dynamics and a fixed actuation channel, where the learnable part enters only through the control input [BRO15, KB59]. For sampled-data swarm control under linear dynamics [CF12, FG02], this suggests that the relevant window-level object is not an averaged state velocity, but a control-space quantity associated with the finite-horizon response of the system. A natural choice is the minimum-energy control over each interval. Indeed, for a controllable linear system, the finite-horizon steering problem admits a closed-form minimum-energy control, determined by a coefficient chosen once per interval. Accordingly, the learned object should be aligned with this sampled-data implementation. Figure 1: Swarm distribution evolves from an initial law ρ0 _0 at t0=0t_0=0 to a target law ρ1 _1 at tK=1t_K=1 under the linear dynamics z˙=Az+Bu z=Az+Bu. The proposed model learns an interval coefficient ck=cθ(zk,tk,tk+1)c_k=c_θ(z_k,t_k,t_k+1), which generates the finite-horizon minimum-energy control over [tk,tk+1][t_k,t_k+1]. To this end, we propose a framework for sampled-data swarm control under linear time-invariant dynamics in the spirit of MeanFlow. Specifically, on each interval, we learn the coefficient of the finite-horizon minimum-energy control. The resulting supervision target is available explicitly from sampled bridge states through the controllability Gramian. In implementation, the learned controller is applied through sampled-data window updates, so the prescribed dynamics and actuation map are preserved by construction. In this way, we place MeanFlow in a control framework and provide a scalable approach to few-step swarm steering under sampled-data linear dynamics [CF12, FG02]. The organization of the rest of the paper is as follows. Section 2 recalls flow matching and MeanFlow. Section 3 formulates the sampled-data MeanFlow problem. Section 4 presents learning and implementation. Section 5 provides numerical illustrations, and Section 6 concludes the paper. 2 Recap: Flow Matching and MeanFlow We begin by recalling the learning framework underlying flow matching and MeanFlow. Both methods are built on bridge-based supervision. Rather than learning directly from a full trajectory distribution, one starts from two endpoint distributions and specifies how an intermediate state is generated between paired samples from these endpoints. This bridge construction converts the generative problem into a supervised one, because each intermediate point comes equipped with a target quantity induced by the bridge. In flow matching, this target is instantaneous and takes the form of a velocity at a single time. MeanFlow modifies this viewpoint by replacing the instantaneous object with one associated with a finite time interval, thereby aligning the learned quantity more closely with coarse-time execution. Let ρdata _data and ρbase _base be two distributions on ℝdR^d, and let z0∼ρdataz_0 _data and z1∼ρbasez_1 _base. A bridge specifies how intermediate states are generated from these endpoint samples. Formally, it is a family of maps ℬtt∈[0,1]\B_t\_t∈[0,1] that induces a differentiable trajectory zt=ℬt(z0,z1),z_t=B_t(z_0,z_1), with instantaneous velocity vt=z˙t=dtℬt(z0,z1).v_t= z_t= ddtB_t(z_0,z_1). In this way, each sampled triple (z0,z1,t)(z_0,z_1,t) determines a supervised pair (zt,vt)(z_t,v_t). A standard example is the linear interpolation zt=(1−t)z0+tz1,z_t=(1-t)z_0+tz_1, with constant velocity vt=z1−z0v_t=z_1-z_0 along the path. Flow matching then seeks to learn a time-dependent vector field from the supervised samples induced by the bridge [LCB+23, MAT+25, YDX25]. Let vθ(z,t)v_θ(z,t) be a parametric velocity field. The standard training objective is minθz0,z1,t‖vθ(zt,t)−vt‖2s.t. zt=ℬt(z0,z1), _θ\;E_z_0,z_1,t \|v_θ(z_t,t)-v_t \|^2 .t. z_t=B_t(z_0,z_1), where the expectation is taken over z0∼ρdataz_0 _data, z1∼ρbasez_1 _base, and t∼(0,1)t (0,1). In this formulation, the learned object is instantaneous, since the supervision specifies the velocity only at the single time t rather than over a finite time interval. MeanFlow starts from the observation that, in practice, the learned dynamics are often executed over a finite time interval rather than in the infinitesimal-step limit [GDB+25a]. It suggests that the quantity to be learned should also be tied to a finite window. In the unstructured setting, MeanFlow therefore replaces the instantaneous velocity target by a window-level object, namely the forward average velocity over the interval [t,r][t,r],111The original MeanFlow is presented in a backward-time convention. Here we adopt a forward-time convention, so the finite-window object is defined on [t,r][t,r] with r>tr>t. u(zt,t,r):=1r−t∫trv(zτ,τ)τ,r>t.u(z_t,t,r):= 1r-t _t^rv(z_τ,τ)\,dτ, r>t. This quantity summarizes the average velocity from the current state ztz_t to the future state zrz_r, and is therefore more naturally aligned with finite-step propagation than the instantaneous velocity v(zt,t)v(z_t,t). For fixed r, differentiating this expression with respect to t yields an identity that relates the window-level object to local quantities along the bridge. This leads to the training target utgt=v(zt,t)+(r−t)[∂zu¯θ(zt,t,r)v(zt,t)+∂tu¯θ(zt,t,r)],u_tgt=v(z_t,t)+(r-t) [ _z u_θ(z_t,t,r)\,v(z_t,t)+ _t u_θ(z_t,t,r) ], which replaces the instantaneous supervision used in standard flow matching with supervision defined over the interval [t,r][t,r]. The corresponding regression objective is minθz0,z1,t,r‖u¯θ(zt,t,r)−sg[](utgt(zt,t,r))‖2, _θ\;E_z_0,z_1,t,r \| u_θ(z_t,t,r)-sg\! [\! ] (u_tgt(z_t,t,r) ) \|^2, where the expectation is taken over endpoint samples together with times t<rt<r, and sg[⋅]sg\! [· ] denotes the stop-gradient operator [GDB+25a]. The essential point is that MeanFlow no longer learns only the local bridge velocity at a single time. Instead, it learns an object associated with a finite interval, which makes the learned dynamics more compatible with coarse-time execution and less sensitive to discretization error over long steps. This interval-based viewpoint is the one we will adapt in the controlled setting. 3 Swarm Control via MeanFlow We consider a large-scale swarm whose configuration at time t is described, at the macroscopic level, by a probability distribution ρt _t on ℝdR^d. The objective is to guide this distribution from a prescribed initial configuration ρ0 _0 to a target configuration ρ1 _1. Rather than designing separate controls for individual agents, we seek a description at the level of the evolving ensemble, where the collective motion is shaped through an underlying controlled dynamics. At the level of a single state, we consider the linear system z˙(τ)=Az(τ)+Bu(τ),τ∈[0,1], z(τ)=Az(τ)+Bu(τ), τ∈[0,1], (1) where A∈ℝd×dA ^d× d and B∈ℝd×mB ^d× m are fixed matrices, and u(τ)∈ℝmu(τ) ^m is the control input. The natural question is the following: given two times t<rt<r and two states zt,zr∈ℝdz_t,z_r ^d, how should one drive the system from ztz_t to zrz_r over the interval [t,r][t,r] while expending the least possible effort? This leads to the classical finite-horizon minimum-energy steering problem minu(⋅)∫tr‖u(τ)‖2τsubject toz˙(τ)=Az(τ)+Bu(τ),z(t)=zt,z(r)=zr. aligned _u(·) & _t^r\|u(τ)\|^2\,dτ\\ subject to & z(τ)=Az(τ)+Bu(τ),\\ &z(t)=z_t, z(r)=z_r. aligned (2) The problem is classical, but here it plays a central conceptual role. It tells us what the natural finite-window object should be in the controlled setting: not an instantaneous velocity, but the quantity that parameterizes the least-effort transfer across the entire interval. The variation-of-constants formula gives zr=Φ(r,t)zt+∫trΦ(r,τ)Bu(τ)τ, z_r= (r,t)z_t+ _t^r (r,τ)B\,u(τ)\,dτ, (3) where Φ(r,t):=eA(r−t) (r,t):=e^A(r-t) denotes the state transition matrix. Assumption 1 (Controllability). The pair (A,B)(A,B) is controllable. Equivalently, for every r>tr>t, the Gramian W(t,r):=∫trΦ(r,τ)BB⊤Φ(r,τ)⊤τ W(t,r):= _t^r (r,τ)B (r,τ) \,dτ (4) is nonsingular. Under Assumption 1, every pair of states (zt,zr)(z_t,z_r) can be connected over [t,r][t,r], and problem (2) admits a unique minimizer. More precisely, the optimal control is of the form u⋆(τ∣t,r)=B⊤Φ(r,τ)⊤c(zt,t,r),τ∈[t,r], u (τ t,r)=B (r,τ) c(z_t,t,r), τ∈[t,r], (5) where, along the bridge, the coefficient c(zt,t,r)∈ℝdc(z_t,t,r) ^d is given by c(zt,t,r)=W(t,r)−1(zr−Φ(r,t)zt). c(z_t,t,r)=W(t,r)^-1 (z_r- (r,t)z_t ). (6) Substituting (5) into (3), one recovers zr=Φ(r,t)zt+W(t,r)c(zt,t,r). z_r= (r,t)z_t+W(t,r)c(z_t,t,r). (7) Thus, the entire optimal transfer over the interval [t,r][t,r] is encoded by the single vector c(zt,t,r)c(z_t,t,r). This coefficient summarizes, in a compressed yet exact form, the steering action required to move from the present state to the future one. It is therefore the natural analogue, in the controlled linear setting, of the window-level object that MeanFlow seeks to learn. We now place this construction in the bridge-based training framework. Let t↦ztt z_t be a differentiable bridge trajectory connecting samples from the endpoint distributions. For a sampled pair of times t<rt<r, the states ztz_t and zrz_r lie on the same bridge trajectory, so the future state zrz_r is determined once the bridge and the forward window are fixed. For this reason, the corresponding finite-horizon steering coefficient is naturally associated with the current bridge state and the interval [t,r][t,r], and we denote it by c(zt,t,r)c(z_t,t,r). It satisfies W(t,r)c(zt,t,r)=zr−Φ(r,t)zt. W(t,r)c(z_t,t,r)=z_r- (r,t)z_t. (8) This is the finite-horizon quantity we seek to learn. The coefficient c(zt,t,r)c(z_t,t,r) also admits an integral representation that reveals its dynamical meaning. If the bridge evolves according to z˙τ=v(zτ,τ), z_τ=v(z_τ,τ), then, by combining (3) and (8), we obtain W(t,r)c(zt,t,r)=∫trΦ(r,τ)(z˙τ−Azτ)τ. W(t,r)c(z_t,t,r)= _t^r (r,τ) ( z_τ-Az_τ )\,dτ. (9) If, moreover, the bridge dynamics are expressed through the input channel as z˙τ=Azτ+Bv(zτ,τ), z_τ=Az_τ+Bv(z_τ,τ), then the identity reduces to W(t,r)c(zt,t,r)=∫trΦ(r,τ)Bv(zτ,τ)τ. W(t,r)c(z_t,t,r)= _t^r (r,τ)B\,v(z_τ,τ)\,dτ. (10) These relations show that the coefficient does not merely describe an endpoint discrepancy. It gathers, over the entire interval, the cumulative action needed to produce the transfer. In this sense, it is the proper finite-window object for the controlled problem, and the one that naturally takes the place of the averaged velocity in the original MeanFlow construction. While c(zt,t,r)c(z_t,t,r) is defined through a finite-horizon relation, it also satisfies a local identity along the bridge. This is what allows the interval coefficient to be learned from local information. Starting from (10), we differentiate with respect to the left endpoint t. Since dtW(t,r)=−Φ(r,t)BB⊤Φ(r,t)⊤, ddtW(t,r)=-\, (r,t)B (r,t) , we obtain W(t,r)dtc(zt,t,r)−Φ(r,t)BB⊤Φ(r,t)⊤c(zt,t,r)=−Φ(r,t)Bv(zt,t). aligned W(t,r) ddtc(z_t,t,r)&- (r,t)B (r,t) c(z_t,t,r)\\ &=- (r,t)B\,v(z_t,t). aligned (11) This relation has a clear interpretation. Although c(zt,t,r)c(z_t,t,r) summarizes the steering action over the entire interval [t,r][t,r], its evolution with respect to the current time t is governed by the local bridge velocity at ztz_t. In this way, the finite-window object remains accessible through local supervision. Motivated by this identity, we introduce a parametric field cθ(zt,t,r)∈ℝd,c_θ(z_t,t,r) ^d, which is intended to approximate the forward finite-horizon steering coefficient along sampled bridge trajectories. The training objective is chosen so that cθc_θ satisfies (11) in the least-squares sense. Problem 1. Consider the linear system (1) with fixed matrices A and B. Under Assumption 1, learn a parametric field cθ(zt,t,r)c_θ(z_t,t,r) by minimizing222In implementation, the target term is detached through a stop-gradient operator, exactly as in MeanFlow, so that the optimization is carried out with a stable one-sided regression objective. ℒ(θ):=z0,z1,t,r‖Rθ(zt,t,r)‖2, (θ):=E_z_0,z_1,t,r \|R_θ(z_t,t,r) \|^2, where the residual is defined by Rθ(zt,t,r) R_θ(z_t,t,r) :=Φ(r,t)BB⊤Φ(r,t)⊤cθ(zt,t,r) := (r,t)B (r,t) c_θ(z_t,t,r) −sg[(W(t,r)dtcθ(zt,t,r)+Φ(r,t)Bv(zt,t))], :=-sg\! [ (W(t,r) ddtc_θ(z_t,t,r)+ (r,t)B\,v(z_t,t) ) ], and (zt,zr)(z_t,z_r) are sampled along the training bridge with t<rt<r. Problem 1 places the learning problem directly in coefficient space. The object being learned is not an instantaneous velocity in state space, but the forward finite-horizon coefficient that parameterizes the minimum-energy steering action over the interval [t,r][t,r]. In this way, the model learns the natural MeanFlow object associated with the linear controlled dynamics. The next observation shows that this construction is fully consistent with the original MeanFlow viewpoint, and that it reduces, in the appropriate limiting case, to the familiar instantaneous supervision of flow matching. Lemma 1. When A=0A=0 and B=IB=I, Problem 1 reduces to the forward MeanFlow formulation. Moreover, if the bridge trajectory is differentiable, then in the limit r→t+r→ t^+ the finite-window target converges to the standard flow matching target [MAT+25]. Proof. Consider first the case A=0A=0 and B=IB=I. Then Φ(r,t)=I (r,t)=I and W(t,r)=∫trIτ=(r−t)I.W(t,r)= _t^rI\,dτ=(r-t)I. Accordingly, the defining relation (6) becomes (r−t)c(zt,t,r)=zr−zt,(r-t)c(z_t,t,r)=z_r-z_t, and hence c(zt,t,r)=zr−ztr−t.c(z_t,t,r)= z_r-z_tr-t. This is precisely the forward MeanFlow target over the interval [t,r][t,r]. Assume in addition that the bridge trajectory is differentiable. Then zr−ztr−t→z˙tas r→t+. z_r-z_tr-t→ z_t r→ t^+. Therefore, limr→t+c(zt,t,r)=z˙t. _r→ t^+c(z_t,t,r)= z_t. Thus, in the infinitesimal-window limit, the finite-horizon target reduces to the instantaneous bridge velocity, which is exactly the supervision target used in flow matching. ∎ Notably, the formulation is not restricted to first-order swarm kinematics. The state variable z may include higher-order components. For instance, in a second-order model one may write z=[xv]⊤,z= bmatrixx&v bmatrix , so that the control enters through an acceleration channel. In that case, c(zt,t,r)c(z_t,t,r) represents the coefficient of the corresponding finite-horizon minimum-energy control over the interval [t,r][t,r]. More broadly, the same construction extends without change to higher-order linear state-space models, as well as to bridge trajectories induced by optimal transport couplings between the initial and target swarm distributions [TFM+24]. 3.1 Sampled-Data Interpretation The coefficient learned by the model has a direct operational meaning. Once a current state ztz_t and a future time r>tr>t are given, the field cθ(zt,t,r)c_θ(z_t,t,r) determines the control profile to be applied over the entire interval [t,r][t,r]. Specifically, the corresponding control signal is u(τ)=B⊤Φ(r,τ)⊤cθ(zt,t,r),τ∈[t,r], u(τ)=B (r,τ) c_θ(z_t,t,r), τ∈[t,r], (12) and the induced state update is zr=Φ(r,t)zt+W(t,r)cθ(zt,t,r). z_r= (r,t)z_t+W(t,r)c_θ(z_t,t,r). (13) Thus, the learned coefficient is not merely a training target. It is precisely the quantity consumed by the finite-horizon sampled-data update over the window [t,r][t,r]. The exact interval coefficient also possesses a natural composition property across adjacent windows. This shows that the finite-horizon steering action is consistent under temporal subdivision. Lemma 2 (Additivity). Let t<s<rt<s<r. Then along any trajectory of (1), W(t,r)c(zt,t,r)=Φ(r,s)W(t,s)c(zt,t,s)+W(s,r)c(zs,s,r). W(t,r)c(z_t,t,r)= (r,s)W(t,s)c(z_t,t,s)+W(s,r)c(z_s,s,r). Proof. By definition, W(t,r)c(zt,t,r)=zr−Φ(r,t)zt.W(t,r)c(z_t,t,r)=z_r- (r,t)z_t. Introduce the intermediate state zsz_s, and use the semigroup property Φ(r,t)=Φ(r,s)Φ(s,t) (r,t)= (r,s) (s,t) to write zr−Φ(r,t)zt z_r- (r,t)z_t =zr−Φ(r,s)zs+Φ(r,s)(zs−Φ(s,t)zt). =z_r- (r,s)z_s+ (r,s) (z_s- (s,t)z_t ). Applying the definition of the interval coefficient on the subintervals [t,s][t,s] and [s,r][s,r] yields zr−Φ(r,t)zt z_r- (r,t)z_t =W(s,r)c(zs,s,r)+Φ(r,s)W(t,s)c(zt,t,s), =W(s,r)c(z_s,s,r)+ (r,s)W(t,s)c(z_t,t,s), which proves the claim. ∎ Lemma 2 shows that the steering action over a long interval may be decomposed into two successive shorter transfers without losing consistency with the dynamics. In this sense, the coefficient provides a dynamically coherent window-level description of the controlled evolution. When cθc_θ does not exactly coincide with the true interval coefficient, the discrepancy appears directly at the level of the sampled-data update. Indeed, one may write zr=Φ(r,t)zt+W(t,r)cθ(zt,t,r)+η(t,r), z_r= (r,t)z_t+W(t,r)c_θ(z_t,t,r)+η(t,r), (14) where η(t,r)=W(t,r)(c(zt,t,r)−cθ(zt,t,r)).η(t,r)=W(t,r) (c(z_t,t,r)-c_θ(z_t,t,r) ). Thus, the state-space error over the interval is exactly the coefficient approximation error through the Gramian. In particular, the residual vanishes when the learned field matches the exact finite-horizon steering coefficient. 4 Control Learning and Implementation We now illustrate how the learned interval-coefficient field is used in practice. The roles of training and sampling are different. During training, a differentiable bridge provides endpoint states from which the finite-horizon steering coefficient is constructed. During sampling, the bridge is no longer used. One evaluates the learned field on each time window and applies the resulting finite-horizon control through the corresponding sampled-data update. 4.1 Interval Control Coefficient Learning Training uses only bridge information over sampled forward windows. For each sampled pair (t,r)(t,r) with t<rt<r, the bridge provides the states ztz_t and zrz_r, together with the local bridge velocity at time t. The network does not learn an instantaneous control signal directly. Instead, it learns the coefficient that determines the minimum-energy control over the entire interval [t,r][t,r]. Accordingly, supervision is imposed through the local differential identity satisfied by the forward finite-horizon steering coefficient, rather than by direct regression onto an explicit teacher coefficient. For training, we use the minimum-energy bridge induced by the linear dynamics between sampled endpoint states z0∼ρ0z_0 _0 and z1∼ρ1z_1 _1. Specifically, for τ∈[0,1]τ∈[0,1], the bridge is given by zτ=Φ(τ,0)z0+W(0,τ)Φ(1,τ)⊤W(0,1)−1(z1−Φ(1,0)z0). z_τ= (τ,0)z_0+W(0,τ) (1,τ) W(0,1)^-1 (z_1- (1,0)z_0 ). (15) This trajectory is the finite-horizon minimum-energy path connecting z0z_0 and z1z_1 under the linear dynamics. Given a sampled forward window [t,r]⊂[0,1][t,r]⊂[0,1], the corresponding bridge states ztz_t and zrz_r are obtained by evaluating (15) at the times t and r. Differentiating (15), or equivalently using the bridge dynamics induced by the minimum-energy control, yields z˙t=Azt+BB⊤Φ(1,t)⊤W(0,1)−1(z1−Φ(1,0)z0). z_t=Az_t+B (1,t) W(0,1)^-1 (z_1- (1,0)z_0 ). (16) Hence, under the input-channel representation z˙t=Azt+Bv(zt,t), z_t=Az_t+Bv(z_t,t), the corresponding bridge action satisfies Bv(zt,t)=BB⊤Φ(1,t)⊤W(0,1)−1(z1−Φ(1,0)z0). Bv(z_t,t)=B (1,t) W(0,1)^-1 (z_1- (1,0)z_0 ). (17) Algorithm 1 summarizes the resulting training procedure. Algorithm 1 Training the interval-coefficient model cθc_θ 1:Initial and target swarm distributions ρ0 _0 and ρ1 _1, system matrices A and B, parametric interval-coefficient model cθ(z,t,r)∈ℝdc_θ(z,t,r) ^d 2:while not converged do 3: Sample endpoint states and training window z0∼ρ0,z1∼ρ1,t∼(0,1),r∼(t,1)z_0 _0,\;z_1 _1,\;t (0,1),\;r (t,1) 4: Compute the bridge states ztz_t and zrz_r from (15) 5: Compute the bridge action at time t Bv(zt,t)=BB⊤Φ(1,t)⊤W(0,1)−1(z1−Φ(1,0)z0) Bv(z_t,t)=B (1,t) W(0,1)^-1 (z_1- (1,0)z_0 ) 6: Evaluate the predicted coefficient cθ(zt,t,r)c_θ(z_t,t,r) 7: Compute the residual Rθ(zt,t,r)R_θ(z_t,t,r) from the differential identity 8: Update θ by one gradient step on ℒ(θ)L(θ) 9:end while The bridge (15) is used only during training, as a source of finite-window supervision. The network itself learns the coefficient that determines the control action over the entire interval. Since the training signal is imposed through the forward differential identity, the learned object remains window-level even though the loss is local in time. 4.2 Interval Control Implementation After training, the output is the learned coefficient field cθ(zt,t,r)c_θ(z_t,t,r). Starting from a sample drawn from the initial swarm distribution, the swarm state is propagated forward along a discrete time grid. On each interval, the network is queried once to produce a single coefficient, and that coefficient determines the finite-horizon control over the window. The state is then propagated by the corresponding sampled-data update. Algorithm 2 summarizes the resulting swarm evolution. Algorithm 2 Finite-horizon forward swarm propagation 1:Swarm distribution ρ0 _0, system matrices A, B 2:Trained interval-coefficient model cθ(z,t,r)∈ℝdc_θ(z,t,r) ^d 3:Choose a time grid 0=t0<t1<⋯<tK=10=t_0<t_1<·s<t_K=1 4:Draw an initial swarm state sample z0∼ρ0z_0 _0 5:for k=0,…,K−1k=0,…,K-1 do 6: Set the current time window to [tk,tk+1][t_k,t_k+1] 7: Compute predicted coefficient ck=cθ(zk,tk,tk+1)c_k=c_θ(z_k,t_k,t_k+1) 8: Propagate swarm state with finite-horizon update zk+1=Φ(tk+1,tk)zk+W(tk,tk+1)ckz_k+1= (t_k+1,t_k)z_k+W(t_k,t_k+1)c_k 9:end for 10:Return swarm trajectory zkk=0K\z_k\_k=0^K The state update above is equivalent to applying, on each interval [tk,tk+1][t_k,t_k+1], the control signal uk(τ)=B⊤Φ(tk+1,τ)⊤ck,τ∈[tk,tk+1].u_k(τ)=B (t_k+1,τ) c_k, τ∈[t_k,t_k+1]. Thus, the controller is updated only at the sampling instants, while the within-window control is determined analytically by the system matrices and the learned coefficient. Although the learning formulation is derived from bridge supervision, the implemented controller is not open-loop. The learned field cθ(z,t,r)c_θ(z,t,r) depends on the current state z, so the control applied on each interval is recomputed online as ck=cθ(zk,tk,tk+1)c_k=c_θ(z_k,t_k,t_k+1). The resulting implementation is therefore a sampled-data state-feedback controller. Remark 1. Problem 1 also provides a natural degree of robustness. If the swarm state is perturbed, then the control on the next interval changes accordingly through the state dependence of cθc_θ. The same mechanism allows the controller to respond to particle loss or stochastic perturbations during propagation, since the control action is generated from the current state. 5 Numerical Illustrations We present two case studies for large-scale swarm control. Our first case considers planar navigation for large-scale ground robots, while the second concerns spatial maneuvering in three dimensions, representative of aerial or underwater swarms. All numerical experiments are implemented in PyTorch and executed on an NVIDIA Tesla V100-SXM2-32GB GPU. The code to conduct all experiments can be found at https://github.com/dytroshut/Meanflow_Control. 5.1 Two-Dimensional Planar Swarm Navigation We first consider two-dimensional planar swarm navigation. The initial swarm distribution at t=0t=0 forms the initials of the authors’ first names, “AYKJ”, and the target distribution at t=1t=1 forms the authors’ last names, “DCJK”. Figure 2 reports the evolution of the swarm distribution, with snapshots taken at t=0, 0.25, 0.5, 0.75,t=0,\,0.25,\,0.5,\,0.75, and 11. Throughout, we color particles according to their spatial location to visually track deformation and the induced correspondence over time. The first row corresponds to the drift-free case A=0A=0 with B=IB=I. This setting reduces to a standard MeanFlow-like generation model, where the dynamics are purely control-driven and the learned generation is carried out with 1616 steps. In this case, the swarm deforms in a direct manner, and the induced origin-to-destination correspondence is largely aligned with the horizontal direction in the plotted configuration. The second row introduces a rotational drift. We set B=IB=I and choose A=[0−ω0],A= bmatrix0&-ω\\ ω&0 bmatrix, which generates a planar rotation. In our parameter setting, the accumulated drift over the horizon corresponds to a 90∘90 rotation. This change in the underlying dynamics reshapes the entire transport process. While the same endpoint distributions are enforced, the induced origin-to-destination correspondence changes with respect to the dynamics. The colored particles that were matched along a horizontal direction in the drift-free case are now matched along a vertical direction, reflecting the rotation imposed by the drift. This example highlights the central point of our framework: the underlying dynamics alter the geometry of the feasible paths and, consequently, the transport plan between the endpoint distributions. Figure 2: Two-dimensional case: swarm from “AYKJ” to “DCJK”. 5.2 Three-Dimensional Spatial Swarm Maneuvering Finally, we showcase our model on a geometrically more demanding three-dimensional task (Figure 3), where the swarm is transported from an initial pyramid to a target torus. This example is challenging in several ways. The swarm must reorganize its spatial support in ℝ3R^3, carve out a hollow central region to form the torus, and do so through an evolution that remains coherent rather than collapsing into unstable or fragmented motion. In Figure 3 we compare several underlying dynamics. The first row uses the standard MeanFlow setting (drift-free dynamics). The second and third rows introduce rotational drift restricted to the x–y plane, illustrating how planar rotation reshapes the intermediate evolution with the same endpoint distributions. The fourth row applies a combined rotational drift in three dimensions. In all cases, the controller is implemented using 1616 steps. In the drift-free setting and in the x–y rotational cases, the tracked particles evolve largely within the expected planes, and the resulting matching retains a relatively direct geometric structure. In contrast, when the rotational drift acts in all three dimensions, the drift fundamentally alters the feasible paths. This leads to a noticeably different intermediate geometry and, consequently, a different final matching pattern. 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