Paper deep dive
Multi-Agent Flow Matching with Decoupled Generative Guidance
Ruoyu Lin, Magnus Egerstedt, Fabio Pasqualetti
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 91%
Last extracted: 9/30/2026, 6:30:44 AM
Summary
The paper introduces DeGG-Flow, a framework for multi-agent flow matching that uses decoupled generative guidance to satisfy hard constraints. It models the generative process as a control-affine dynamical system and defines two classes of requirements: shared (dependent on multiple agents) and private (dependent on neighbors). The framework provides formal feasibility and finite-horizon convergence guarantees for both classes and derives a Wasserstein bound for distributional deviation. It is demonstrated on multi-robot collaboration and multi-object scene generation, showing it can generate objects satisfying hard requirements without retraining.
Entities (10)
Relation Signals (9)
DeGG-Flow → uses → Flow Matching
confidence 95% · DeGG-Flow, a general framework for multi-agent flow matching
DeGG-Flow → demonstratedon → Multi-Object Scene Generation
confidence 93% · and on multi-object scene generation with affordance requirements.
DeGG-Flow → demonstratedon → Multi-Robot Collaboration
confidence 93% · We demonstrate DeGG-Flow on multi-robot collaboration
DeGG-Flow → models → Control-Affine Dynamical System
confidence 92% · By representing the generative process as a control-affine dynamical system
DeGG-Flow → handles → Shared Requirements
confidence 90% · DeGG-Flow handles two classes of requirements: shared requirements... and private requirements
DeGG-Flow → handles → Private Requirements
confidence 90% · DeGG-Flow handles two classes of requirements: ... private requirements
DeGG-Flow → provides → Wasserstein Bound
confidence 88% · We further derive a Wasserstein bound that characterizes the distributional deviation
DeGG-Flow → uses → Graph Neural Network
confidence 87% · represented as a message passing graph neural network (GNN)
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:Generative modeling is widely used for producing diverse objects from complex, multimodal distributions. However, its expressivity does not, in general, come with formal guarantees that the generated objects satisfy hard constraints or requirements. In multi-agent generation, this problem becomes more challenging because a hard requirement can depend on multiple agents, while each agent may need to determine its own guidance input without relying on the simultaneously computed guidance inputs of other agents. To this end, we introduce DeGG-Flow, a general framework for multi-agent flow matching with decoupled generative guidance. By representing the generative process as a control-affine dynamical system, we develop guidance conditions for two classes of coupled requirements: shared requirements whose satisfaction depends on multiple agents together, and private requirements associated with each individual agent dependent on its neighbors. For both classes, we establish feasibility conditions and finite-horizon convergence guarantees. We further derive a Wasserstein bound that characterizes the distributional deviation induced by the guidance. We demonstrate DeGG-Flow on multi-robot collaboration for crossing a spatial gap by reconfiguring the environment, and on multi-object scene generation with affordance requirements. Across both applications, DeGG-Flow directly generates objects that satisfy all corresponding hard requirements, including at team sizes unseen during training.
Tags
Links
- Source: https://arxiv.org/abs/2609.38133v1
- Canonical: https://arxiv.org/abs/2609.38133v1
Trouble viewing inline? Open PDF directly →
Full Text
106,666 characters extracted from source content.
Expand or collapse full text
Multi-Agent Flow Matching with Decoupled Generative Guidance Ruoyu Lin Magnus Egerstedt Fabio Pasqualetti Abstract Generative modeling is widely used for producing diverse objects from complex, multimodal distributions. However, its expressivity does not, in general, come with formal guarantees that the generated objects satisfy hard constraints or requirements. In multi-agent generation, this problem becomes more challenging because a hard requirement can depend on multiple agents, while each agent may need to determine its own guidance input without relying on the simultaneously computed guidance inputs of other agents. To this end, we introduce DeGG-Flow, a general framework for multi-agent flow matching with decoupled generative guidance. By representing the generative process as a control-affine dynamical system, we develop guidance conditions for two classes of coupled requirements: shared requirements whose satisfaction depends on multiple agents together, and private requirements associated with each individual agent dependent on its neighbors. For both classes, we establish feasibility conditions and finite-horizon convergence guarantees. We further derive a Wasserstein bound that characterizes the distributional deviation induced by the guidance. We demonstrate DeGG-Flow on multi-robot collaboration for crossing a spatial gap by reconfiguring the environment, and on multi-object scene generation with affordance requirements. Across both applications, DeGG-Flow directly generates objects that satisfy all corresponding hard requirements, including at team sizes unseen during training. 11footnotetext: Ruoyu Lin and Fabio Pasqualetti are with the Department of Electrical Engineering and Computer Science, University of California, Irvine, Irvine, CA 92697 USA. Email: rlin10,fabiopas@uci.edu 22footnotetext: Magnus Egerstedt is with the University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 USA. Email: magnus@unc.edu 1 Introduction Generative models have achieved remarkable success across a wide range of domains, from language and vision to robotics and scientific discovery (Brown et al., 2020; Rombach et al., 2022; Chi et al., 2025; Lipman et al., 2022; Albergo et al., 2025; Janner et al., 2022; Tang et al., 2024). A core strength of these models is their ability to characterize complex, multimodal distributions and generate diverse objects from them. However, such expressivity, in general, does not imply that every generated object satisfies desired hard constraints or requirements. For example, a generated motion may violate safety requirements, a generated molecular structure may violate physical or chemical constraints, and new requirements introduced after training may not be represented in the training data. This gap is especially important when a learned generative model is already useful but new hard requirements arise after training. In such cases, it is desirable to impose constraints at generation time without collecting new data or retraining the model. Guidance in the generative process provides a natural mechanism for enforcing constraints (Bansal et al., 2024; Feng et al., 2025). Recent work has made this idea increasingly principled by incorporating control barrier functions (CBF) (Ames et al., 2016) into generative process. SafeDiffuser (Xiao et al., 2025) introduces CBF into the denoising process of diffusion probabilistic model (Ho et al., 2020); SafeFlow (Dai et al., 2025) targets robot motion planning by imposing CBF constraints at the waypoints of a generated trajectory; Gadginmath et al. (2026) formulate constrained sampling through a stochastic differential equation and implement discrete-time CBF through a first-order approximation; and SafeFlowMatcher (Yang et al., 2026) addresses robotic path planning by first generating a candidate path and then applying a separate CBF-based correction. Multi-agent generative modeling changes the structure of such guidance, as hard constraints or requirements may depend jointly on multiple agents. A direct approach is to treat all agents as a single-agent system and compute the guidance together. However, a key advantage of multi-agent systems is that the team need not depend on a single central decision maker, whose failure could affect all agents at once. This motivates decentralized or agent-wise guidance, where each agent determines its own guidance without relying on that of any other agent. Existing multi-agent generative models address different aspects of this problem. MADiff (Zhu et al., 2024) models multimodal multi-agent behavior and supports decentralized execution; Shaoul et al. (2025) combine diffusion models with search-based planning to generate collision-free multi-robot trajectories; Parimi and Williams (2025) decompose multi-arm planning into single-arm trajectory generation and pairwise collision resolution; Liang et al. (2025) integrate constrained projection into the denoising process (Ho et al., 2020) to satisfy collision avoidance and kinematic constraints; and MAC-Flow (Lee et al., 2026) learns joint behaviors with flow matching and distills them into decentralized one-step policies. These works provide different mechanisms for generating, coordinating, and adapting multi-agent behavior. What remains unresolved is how to guarantee a hard requirement that depends on multiple agents when each agent determines only its own guidance. The constrained generative methods discussed earlier do not directly resolve this problem either. Applied to the full multi-agent state, they naturally lead to a joint guidance problem. The challenge is thus to formulate a guidance rule for each agent that it can satisfy, while ensuring that these agent-level rules together provide provable guarantees on the team-level coupled constraints or requirements. To this end, we introduce DeGG-Flow, a general framework for multi-agent flow matching with decoupled generative guidance. Its central idea is to preserve agent-wise guidance while providing formal guarantees for requirements that can couple multiple agents. DeGG-Flow handles two classes of requirements: shared requirements for which multiple agents collaboratively contribute to their satisfaction, and private requirements associated with each individual agent whose satisfaction depends on its neighbors. The guidance is applied during the generative process, without collecting new training data, retraining the learned generative model, or repairing the generated objects afterward. The distributional deviation induced by the guidance is further shown to be theoretically bounded rather than uncharacterized. Our main contributions are summarized as follows. • Agent-wise guidance with team-level requirement. We formulate the generative process of multi-agent flow matching as a control-affine dynamical system and adapt disentangled control (Lin et al., 2026) to decoupled guidance, so that each agent determines only its own guidance input without relying on the simultaneously computed guidance input of any other agent, while retaining formal guarantees for requirements that couple multiple agents. • Feasibility and finite-horizon convergence guarantees. For shared requirements, we introduce an adaptive constraint allocation approach to ensure feasibility of the corresponding decoupled guidance, and propose a time-varying bound that guarantees the generated object reaches a desired set in the generative state space within finite horizon. For private requirements, we also establish feasibility conditions for the corresponding decoupled guidance and propose an approach to guarantee finite-horizon convergence. • Bounded distributional deviation under guidance. We derive a Wasserstein bound that characterizes how far the guided joint generative distribution can deviate from the unguided one, and empirically evaluate the resulting distributional deviation in experiments. • Applications with shared and private requirements. We demonstrate DeGG-Flow on multi-robot collaborative policy generation with shared coupled requirements for crossing a spatial gap by reconfiguring the environment, and on multi-object scene generation with private coupled requirements, where each object’s affordance requirements, i.e., its access and use requirements, depend on neighboring objects. In both tasks, DeGG-Flow directly generates objects that satisfy all hard requirements without post-generation correction, including for numbers of agents beyond those used during training. 2 Multi-Agent Flow Matching Guided by Disentangled Control 2.1 Multi-agent flow matching Consider a team of N∈ℤ+N ^+ agents, indexed by i∈≔1,…,Ni \1,…,N\, whose joint generative process produces an output y∈Ny _N for a given task, where NY_N denotes the corresponding output space. Let χ∈Nχ _N denote the generation condition, where NC_N is the corresponding condition space. The condition contains information such as initial states, agent or object properties, environment information, or task specifications. The output y depends on specific applications. For example, in Section 4.1, y represents a motion policy for N robots, and in Section 4.2, y represents an arrangement of N objects. Instead of generating y directly, each agent i has a generative state zi(τ)∈⊆ℝdzz_i(τ) ^d_z at generative time τ∈[0,1]τ∈[0,1], which is different from physical time t∈ℝ≥0t _≥ 0. Denote the joint generative state as (τ)≔[z1(τ)⊤,…,zN(τ)⊤]⊤∈N⊆ℝNdzz(τ) [z_1(τ) ,…,z_N(τ) ] ^N ^Nd_z. The task-specific decoder and encoder map between the joint generative state space and the output space are denoted as N _N :N×N→N,y=N(|χ), :Z^N×C_N _N, y=D_N(z\,|\,χ), (1) ℰN _N :N×N→N,=ℰN(y|χ). :Y_N×C_N ^N, =E_N(y\,|\,χ). (2) For a training data y, ℰN(y|χ)E_N(y\,|\,χ) per (2) gives its target generative state for flow matching, and N((1)|χ)D_N(z(1)\,|\,χ) per (1) represents the final generated object. Given N and χ, let pdata(|χ)p_data(z\,|\,χ) denote the joint conditional distribution of the encoded training data in NZ^N, and let p0()p_0(x) denote a simple initial distribution in NZ^N from which samples can be drawn efficiently. Generation starts from (0)∼p0z(0) p_0 and evolves over τ∈[0,1]τ∈[0,1] governed by the dynamical system ˙=f(τ,|χ). z=f(τ,z\,|\,χ). The objective of flow matching (Lipman et al., 2022) is to learn a vector field whose flow transports p0p_0 to pdata(|χ)p_data(z\,|\,χ). To learn such a vector field, for each target ∈Nz ^N, consider a conditional probability path p(τ,|,χ)p(τ,x\,|\,z,χ) satisfying p(0,|,χ)=p0()p(0,x\,|\,z,χ)=p_0(x) and p(1,|,χ)=δ(−)p(1,x\,|\,z,χ)=δ(x-z), where ∈Nx ^N is a realization of the intermediate random variable (τ)X(τ) and δ(−)δ(x-z) is the Dirac delta concentrated at z. Let v(τ,|,χ)∈ℝNdzv(τ,x\,|\,z,χ) ^Nd_z denote a conditional time-varying vector field that generates this path, then the marginal probability path is p(τ,|χ)=∫Np(τ,|,χ)pdata(|χ)dp(τ,x\,|\,χ)= _Z^Np(τ,x\,|\,z,χ)p_data(z\,|\,χ)\,dz, and the corresponding marginal vector field is f(τ,|χ)≔∫Nv(τ,|,χ)p(τ,|,χ)pdata(|χ)p(τ,|χ)d.f(τ,x\,|\,χ) _Z^N\!v(τ,x\,|\,z,χ)\, p(τ,x\,|\,z,χ)p_data(z\,|\,χ)p(τ,x\,|\,χ)\,dz. (3) Since (3) depends on the unknown data distribution, in this paper, (3) is represented as a message passing graph neural network (GNN) (Gilmer et al., 2017) fθ:[0,1]×N×N→ℝNdz,f^θ:[0,1]×Z^N×C_N ^Nd_z, (4) which we refer to as the nominal vector field. The same neural network parameters θ are shared across all agents. Each agent computes its output from its own state and messages aggregated from neighboring agents so that the GNN (4) is permutation equivariant. Denote fθ=[(f1θ)⊤,…,(fNθ)⊤]⊤f^θ=[(f_1^θ) ,…,(f_N^θ) ] , where fiθ∈ℝdzf_i^θ ^d_z is the nominal vector field of agent i, which is trained using the multi-agent conditional flow matching loss ℒMAC(θ|χ)≔[ _MAC(θ\,|\,χ) [ 1N∑i=1N∥fiθ(τ,(τ)|χ) 1N _i=1^N \|f_i^θ(τ,X(τ)\,|\,χ) (5) −vi(τ,(τ)|,χ)∥2], -v_i(τ,X(τ)\,|\,Z,χ) \|^2 ], in which the expectation is taken over τ∼Unif[0,1]τ [0,1], the encoded data random variable ∼pdata(|χ)Z p_data(z\,|\,χ), and the intermediate state random variable (τ)∼p(τ,|,χ)X(τ) p(τ,x\,|\,Z,χ) along the conditional probability path. After training, fθf^θ is the nominal vector field governing the generative dynamics. We next introduce a guidance mechanism modifying such dynamics. 2.2 Disentangled control guidance The nominal vector field fθf^θ transports samples from a simple initial distribution toward a complex, multimodal distribution that approximates the encoded data distribution. However, the resulting generated objects are not, in general, guaranteed to satisfy hard constraints or requirements. To this end, we introduce a guidance input to the dynamics of each agent in the generative process. Let ui(τ)∈ℝmiu_i(τ) ^m_i be the guidance input of agent i, and gi:[0,1]×N×N→ℝdz×mig_i:[0,1]×Z^N×C_N ^d_z× m_i be the map determining how the guidance input affects the generative dynamics. Then, the guided generative dynamics are in the control-affine form z˙i=fiθ(τ,|χ)+gi(τ,|χ)ui,∀i∈. z_i=f_i^θ(τ,z\,|\,χ)+g_i(τ,z\,|\,χ)u_i, ∀ i . (6) We represent the hard requirement by a C1C^1 (i.e., continuously differentiable) function, and consider two classes of decoupled guidance by adapting disentangled control (Lin et al., 2026) from multi-agent control in physical space into the generative process as a form of decoupled guidance, i.e., guidance for shared-entangled requirements (SE guidance) and guidance for private-entangled requirements (PE guidance), to enable agent i to compute uiu_i without access to the simultaneously computed guidance input uju_j, ∀j∈∖i∀ j \!\i\. SE guidance. Let FSE=(,SE,ℐSE)G_F^SE=(N,A^SE,I^SE) be a factor graph (Kschischang et al., 2001), where SEA^SE is a finite set of factors and ℐSE⊆×SEI^SE ×A^SE is the incidence relation. For each factor a∈SEa ^SE, define the incident agent set as aSE≔i∈:(i,a)∈ℐSE≠∅S_a^SE \i :(i,a) ^SE\≠ , and let aSEz_S_a^SE denote the corresponding subvector of the joint generative state. Let VaSE:[0,1]×∏i∈aSE×N→ℝV_a^SE:[0,1]× _i _a^SEZ×C_N be C1C^1. The guidance input uiu_i of agent i∈aSEi _a^SE is such that ∇ziVaSE(τ,aSE|χ)⊤(fiθ(τ,|χ)CLOSE _z_iV_a^SE(τ,z_S_a^SE\,|\,χ) (f_i^θ(τ,z\,|\,χ) (7) OPEN+gi(τ,|χ)ui)+ζa,iSE(τ,|χ)≤0, +g_i(τ,z\,|\,χ)u_i )+ _a,i^SE(τ,z\,|\,χ)≤ 0, in which ∑i∈aSEζa,iSE(τ,|χ) _i _a^SE _a,i^SE(τ,z\,|\,χ) =∂τVaSE(τ,aSE|χ) = _τV_a^SE(τ,z_S_a^SE\,|\,χ) (8) +αaSE(VaSE(τ,aSE|χ)), + _a^SE\! (V_a^SE(τ,z_S_a^SE\,|\,χ) ), where αaSE:ℝ→ℝ _a^SE:R is an extended class ∞K_∞ function. PE guidance. For each agent i∈i , let ViPE:[0,1]×N×N→ℝ≥0V_i^PE:[0,1]×Z^N×C_N _≥ 0 be C1C^1. The guidance input uiu_i of agent i is such that ∇ziVPE(τ,|χ)⊤(fiθ(τ,|χ)+gi(τ,|χ)ui) _z_iV^PE(τ,z\,|\,χ) (f_i^θ(τ,z\,|\,χ)+g_i(τ,z\,|\,χ)u_i ) (9) +ζiPE(τ,|χ)≤0,∀i∈, + _i^PE(τ,z\,|\,χ)≤ 0, ∀ i , in which VPE(τ,|χ)≔∑i∈ViPE(τ,|χ)V^PE(τ,z\,|\,χ) _i V_i^PE(τ,z\,|\,χ) and ζiPE(τ,|χ) _i^PE(τ,z\,|\,χ) ≔∂τViPE(τ,|χ) _τV_i^PE(τ,z\,|\,χ) (10) +ωPE(τ)αPE(ViPE(τ,|χ)), +ω^PE(τ)\,α^PE\! (V_i^PE(τ,z\,|\,χ) ), where ωPE:[0,1]→ℝ≥0ω^PE:[0,1] _≥ 0, and αPE:ℝ≥0→ℝ≥0α^PE:R_≥ 0 _≥ 0 is a subadditive class ∞K_∞ function. Intuitively, SE guidance is suitable for scenarios where multiple agents have certain shared goals or requirements, such as the task of multi-robot collaborative policy generation in Section 4.1, while PE guidance is suitable for scenarios where each agent has its own requirement, which can be affected by its neighbors, such as the task of multi-object scene generation in Section 4.2. 3 Adaptive Guidance and Formal Guarantees Section 2 introduces two classes of decoupled guidance for multi-agent generation, under which each agent computes its own guidance input without relying on the simultaneously computed guidance inputs of other agents. Since the generative process evolves over the finite horizon τ∈[0,1]τ∈[0,1], it is necessary to ensure that the final generated object directly satisfies the desired hard requirements. In this section, we propose separate finite-horizon convergence methods for SE and PE guidance. We further characterize the distributional deviation induced by guidance through a Wasserstein bound on the guided and nominal joint generative distributions. 3.1 SE guidance with finite-horizon convergence Let the C1C^1 function qaSE(τ,aSE|χ)q_a^SE(τ,z_S_a^SE\,|\,χ) represent a shared requirement, where qaSE≤0q_a^SE≤ 0 means that the requirement is satisfied. An initial sample (0)∼p0z(0) p_0 need not satisfy this requirement. Thus, we introduce a time-varying upper bound βaSE(τ|χ,(0)) _a^SE(τ\,|\,χ,z(0)) that is initialized from (0)z(0). The bound is chosen to satisfy βaSE(0|χ,(0))≥qaSE(0,aSE(0)|χ), _a^SE(0\,|\,χ,z(0))≥ q_a^SE(0,z_S_a^SE(0)\,|\,χ), (11) and is tightened toward a prescribed terminal value. In particular, choosing βaSE(1|χ,(0))=0 _a^SE(1\,|\,χ,z(0))=0 ensures that the desired hard requirement is satisfied by the end of the generative process. Accordingly, define VaSE(τ,aSE|χ)≔qaSE(τ,aSE|χ)−βaSE(τ|χ,(0)),V_a^SE(τ,z_S_a^SE\,|\,χ) q_a^SE(τ,z_S_a^SE\,|\,χ)- _a^SE(τ\,|\,χ,z(0)), (12) so that VaSE≤0V_a^SE≤ 0 is equivalent to qaSE≤βaSEq_a^SE≤ _a^SE. Since factor a involves multiple agents, ∂τVaSE+αaSE(VaSE) _τV_a^SE+ _a^SE(V_a^SE) can be allocated among the involved agents using wa,iSE≥0w_a,i^SE≥ 0 with ∑i∈aSEwa,iSE=1 _i _a^SEw_a,i^SE=1, i.e., ζa,iSE(τ,|χ) _a,i^SE(τ,z\,|\,χ) ≔wa,iSE(∂τqaSE(τ,aSE|χ)CLOSE w_a,i^SE ( _τq_a^SE(τ,z_S_a^SE\,|\,χ) (13) −β˙aSE(τ|χ,(0)) - β_a^SE(τ\,|\,χ,z(0)) OPEN+αaSE(VaSE(τ,aSE|χ))). + _a^SE\! (V_a^SE(τ,z_S_a^SE\,|\,χ) ) ). Since ∂τVaSE=∂τqaSE−β˙aSE _τV_a^SE= _τq_a^SE- β_a^SE, this construction satisfies (8). The following theorem shows that satisfying the agent-wise constraint of SE guidance keeps the shared requirement below its prescribed bound throughout the generative process. Theorem 3.1 (Finite-horizon convergence for SE guidance). Let qaSEq_a^SE and βaSE _a^SE be C1C^1. Suppose (11) holds, and let VaSEV_a^SE and ζa,iSE _a,i^SE be defined by (12) and (13), with wa,iSE≥0w_a,i^SE≥ 0 and ∑i∈aSEwa,iSE=1 _i _a^SEw_a,i^SE=1. If every agent i∈aSEi _a^SE, along a trajectory of (6), satisfies (7), ∀τ∈[0,1]∀τ∈[0,1], then qaSE(τ,aSE(τ)|χ)≤βaSE(τ|χ,(0)),∀τ∈[0,1].q_a^SE(τ,z_S_a^SE(τ)\,|\,χ)≤ _a^SE(τ\,|\,χ,z(0)), ∀τ∈[0,1]. In particular, if βaSE(1|χ,(0))=0, _a^SE(1\,|\,χ,z(0))=0, then qaSE(1,aSE(1)|χ)≤0,q_a^SE(1,z_S_a^SE(1)\,|\,χ)≤ 0, i.e., the final generated state satisfies the shared requirement. Proof. See Appendix A.1. ∎ To ensure that the constraint (7) is feasible for each agent, we adapt the coefficients wa,iSEw_a,i^SE according to how each agent can affect the shared requirements. Define ℓa,iSE≔gi(τ,|χ)⊤∇ziqaSE(τ,aSE|χ), _a,i^SE g_i(τ,z\,|\,χ) _z_iq_a^SE(τ,z_S_a^SE\,|\,χ), which determines how the guidance input uiu_i affects the shared requirement qaSEq_a^SE. Based on (13), the constraint (7) can be written as (ℓa,iSE)⊤ui+ba,iSE≤0( _a,i^SE) u_i+b_a,i^SE≤ 0 with ba,iSE≔∇ziqaSE(τ,aSE|χ)⊤fiθ(τ,|χ)+ζa,iSE(τ,|χ).b_a,i^SE _z_iq_a^SE(τ,z_S_a^SE\,|\,χ) f_i^θ(τ,z\,|\,χ)+ _a,i^SE(τ,z\,|\,χ). When ba,iSE≤0b_a,i^SE≤ 0, ui=0u_i=0 already satisfies this inequality. When ba,iSE>0b_a,i^SE>0, agent i must apply a guidance input that reduces the corresponding requirement. When two SE factors are simultaneously active at agent i, their local guidance conditions may compete for the same input uiu_i. To address this issue, let a∈1,2a∈\1,2\ and ℓ^a,iSE≔ℓa,iSE/‖ℓa,iSE‖ _a,i^SE _a,i^SE/\| _a,i^SE\| if ℓa,iSE≠0 _a,i^SE≠ 0 and ℓ^a,iSE≔0 _a,i^SE 0 otherwise. Then, ηiSE≔(ℓ^1,iSE)⊤ℓ^2,iSE _i^SE ( _1,i^SE) _2,i^SE characterizes how aligned the effects of uiu_i on the two active requirements is. For δ∈(0,1)δ∈(0,1), let ϕδ:[−1,1]→[−1,0] _δ:[-1,1]→[-1,0] be a C2C^2 function satisfying ϕδ(η)=η _δ(η)=η for η≤−δη≤-δ, ϕδ(η)=0 _δ(η)=0 for η≥δη≥δ, and ϕδ(η)≤minη,0 _δ(η)≤ \η,0\ for η∈[−1,1]η∈[-1,1]. Then, we define the effective directions d1,iSE≔ℓ1,iSE−‖ℓ1,iSE‖ϕδ(ηiSE)ℓ^2,iSEd_1,i^SE _1,i^SE-\| _1,i^SE\| _δ( _i^SE) _2,i^SE and d2,iSE≔ℓ2,iSE−‖ℓ2,iSE‖ϕδ(ηiSE)ℓ^1,iSE.d_2,i^SE _2,i^SE-\| _2,i^SE\| _δ( _i^SE) _1,i^SE. When only one SE factor is active at agent i, we set da,iSE≔ℓa,iSEd_a,i^SE _a,i^SE. The modification is inactive when the two effective directions are sufficiently aligned and increases as they become opposed. As such, for each active SE factor, we choose the adaptive constraint allocation coefficient in (13) as wa,iSE≔‖da,iSE‖2∑j∈aSE‖da,jSE‖2.w_a,i^SE \|d_a,i^SE\|^2 _j _a^SE\|d_a,j^SE\|^2. (14) Then, the following proposition establishes the existence of guidance inputs such that (7) is satisfied. Proposition 3.2 (Adaptive SE guidance). If each agent is incident to at most two simultaneously active SE factors, and for every active SE factor the denominator of (14) is nonzero and da,iSE≠0d_a,i^SE≠ 0 when ba,iSE>0b_a,i^SE>0, then, there exists a guidance input ui(τ)u_i(τ) satisfying (7), ∀τ∈[0,1]∀τ∈[0,1]. Such a guidance input can be obtained from minui∈ℝmiui⊤Hiui _u_i ^m_i u_i H_iu_i (15) s.t.(ℓa,iSE)⊤ui+ba,iSE≤0,∀a∈iSE, .t. \;\,( _a,i^SE) u_i+b_a,i^SE≤ 0, ∀ a _i^SE, which admits a unique minimizer when Hi≻0H_i 0, where iSE≔a∈SE:i∈aSEA_i^SE \a ^SE:i _a^SE\. Proof. See Appendix A.2. ∎ 3.2 PE guidance with finite-horizon convergence Next, we propose a method to ensure finite-horizon convergence for PE guidance. Recall that, unlike the real-valued VaSE∈ℝV_a^SE in the SE guidance scenario, here each agent has a nonnegative ViPE∈ℝ≥0V_i^PE _≥ 0 representing the violation of its private requirement that can be dependent on its neighbors. Let ∂τViPE(τ,|χ)=0 _τV_i^PE(τ,z\,|\,χ)=0 and choose the total violation to be VPE(|χ)=∑i∈ViPE(|χ),V^PE(z\,|\,χ)= _i V_i^PE(z\,|\,χ), so VPE=0V^PE=0 means that all private requirements are satisfied. We then choose αPE(s)≔cPEsρPE,α^PE(s) c^PEs^ρ^PE, (16) where cPE∈ℝ>0c^PE _>0 and ρPE∈(0,1)ρ^PE∈(0,1). The choice of 0<ρPE<10<ρ^PE<1 enables finite-horizon convergence, and ωPE(τ)ω^PE(τ) in (10) modulates the required decrease of VPEV^PE over τ, as shown in the following theorem. Theorem 3.3 (Finite-horizon convergence for PE guidance). Let αPEα^PE be given by (16). If every agent, along a trajectory of (6), satisfies (9), ∀τ∈[0,1]∀τ∈[0,1], then VPE((τ)|χ) V^PE(z(τ)\,|\,χ) ≤max0,(VPE((0)|χ))1−ρPE ≤ \0,\; (V^PE(z(0)\,|\,χ) )^1-ρ^PE −cPE(1−ρPE)∫0τωPE(s)ds11−ρPE. -c^PE(1-ρ^PE) _0^τ\!ω^PE(s)\,ds \ 11-ρ^PE. In particular, if cPE(1−ρPE)∫01ωPE(s)s≥(VPE((0)|χ))1−ρPE,c^PE(1-ρ^PE) _0^1\!ω^PE(s)\,ds≥ (V^PE(z(0)\,|\,χ) )^1-ρ^PE, then VPE((1)|χ)=0,V^PE(z(1)\,|\,χ)=0, i.e., the final generated state satisfies all private requirements. Proof. See Appendix A.3. ∎ Unlike SE guidance, PE guidance assigns each agent a single inequality constraint. Define ℓiPE≔gi(τ,|χ)⊤∇ziVPE(|χ) _i^PE g_i(τ,z\,|\,χ) _z_iV^PE(z\,|\,χ) and biPE b_i^PE ≔∇ziVPE(|χ)⊤fiθ(τ,|χ) _z_iV^PE(z\,|\,χ) f_i^θ(τ,z\,|\,χ) +cPEωPE(τ)(ViPE(|χ))ρPE. +c^PEω^PE(τ) (V_i^PE(z\,|\,χ) )^ρ^PE. Then the constraint (9) becomes (ℓiPE)⊤ui+biPE≤0( _i^PE) u_i+b_i^PE≤ 0. If biPE≤0b_i^PE≤ 0, the nominal input ui=0u_i=0 already satisfies the constraint. If biPE>0b_i^PE>0, a feasible guidance input exists when ℓiPE≠0 _i^PE≠ 0. This gives the following feasibility result. Corollary 3.4 (Feasibility of PE guidance). Per Theorem 3.3, if biPE≤0b_i^PE≤ 0, or if biPE>0b_i^PE>0 and ℓiPE≠0 _i^PE≠ 0, then there exists a guidance input ui(τ)u_i(τ) satisfying (9), ∀τ∈[0,1]∀τ∈[0,1]. Such a guidance input can be obtained from minui∈ℝmiui⊤Hiui _u_i ^m_i u_i H_iu_i (17) s.t.(ℓiPE)⊤ui+biPE≤0, .t. \;\,( _i^PE) u_i+b_i^PE≤ 0, which admits a unique minimizer when Hi≻0H_i 0. Proof. See Appendix A.4. ∎ 3.3 Distributional deviation For generated objects to satisfy certain hard constraints or reach a desired set in the generative space, the nominal vector field is modified by SE or PE guidance. We then establish a Wasserstein bound to characterize the resulting deviation from the nominal joint generative distribution. The bound relates this deviation to the magnitude of the guidance correction, providing a theoretical characterization of such intervention. Specifically, consider nominal and guided joint generative state trajectories initialized from the same random sample, i.e., dτnom(τ)=fθ(τ,nom(τ)|χ) ddτX^nom(τ)=f^θ(τ,X^nom(τ)\,|\,χ) and dτgui(τ)=fθ(τ,gui(τ)|χ)+Γ(τ|ξ) ddτX^gui(τ)=f^θ(τ,X^gui(τ)\,|\,χ)+ (τ\,|\,ξ) with nom(0)=gui(0)=ξ∼p0X^nom(0)=X^gui(0)=ξ p_0, where the joint guidance correction is denoted by Γ(τ|ξ)≔ (τ\,|\,ξ) [(g1(τ,gui(τ)|χ)u1(τ))⊤,…, [(g_1(τ,X^gui(τ)\,|\,χ)u_1(τ)) ,…, (18) (gN(τ,gui(τ)|χ)uN(τ))⊤]⊤. (g_N(τ,X^gui(τ)\,|\,χ)u_N(τ)) ] . Let μ1 _1 and ν1 _1 denote the probability distributions of nom(1)X^nom(1) and gui(1)X^gui(1), respectively. Theorem 3.5 (Wasserstein bound on guidance-induced distributional deviation). Assume μ1 _1 and ν1 _1 have finite second moments and, for all τ∈[0,1]τ∈[0,1] and ,∈Ny,z ^N, ‖fθ(τ,|χ)−fθ(τ,|χ)‖≤L(τ)‖−‖, \|f^θ(τ,y\,|\,χ)-f^θ(τ,z\,|\,χ) \|≤ L(τ) \|y-z \|, where L:[0,1]→ℝ≥0L:[0,1] _≥ 0 is integrable. If ∫01∫N‖Γ(s|ξ)‖2p0(ξ)ξs<∞, _0^1\! _Z^N \| (s\,|\,ξ) \|^2p_0(ξ)\,dξ\,ds<∞, where Γ(s|ξ) (s\,|\,ξ) per (18) is the guidance correction corresponding to the initial sample ξ∼p0ξ p_0, then W2(μ1,ν1)≤ W_2( _1, _1)≤ ∫01exp(∫s1L(r)dr) _0^1\! \! ( _s^1\!L(r)\,dr ) (19) ⋅∫N‖Γ(s|ξ)‖2p0(ξ)ξds. ·\, _Z^N\! \| (s\,|\,ξ) \|^2p_0(ξ)\,dξ\;ds. In addition, given χ, assume the decoder satisfies ‖N(|χ)−N(|χ)‖≤LD‖−‖, \|D_N(y\,|\,χ)-D_N(z\,|\,χ) \|≤ L_D \|y-z \|, ∀,∈N ,z ^N, for some LD∈ℝ≥0L_D _≥ 0. Let μ¯1 μ_1 and ν¯1 ν_1 denote the probability distributions of N(nom(1)|χ)D_N(X^nom(1)\,|\,χ) and N(gui(1)|χ)D_N(X^gui(1)\,|\,χ), respectively. Then W2(μ¯1,ν¯1)≤LDW2(μ1,ν1).W_2( μ_1, ν_1)≤ L_DW_2( _1, _1). (20) Proof. See Appendix A.5. ∎ Empirically evaluation of the distributional deviation between the nominal vector field and the guided one is in Appendix C.2. 3.4 Algorithm After establishing the finite-horizon convergence guarantees for SE and PE guidance, Algorithm 1 summarizes the decoupled guidance procedure. The nominal vector field fiθf_i^θ may be provided by a pretrained model, or learned using (5) that can be augmented with task-specific training losses. Such augmentation affects only the learned nominal vector field and is independent of the SE or PE guidance applied during the generative process. Note that the proposed decoupled guidance requires neither retraining nor a particular neural network architecture for the nominal vector field. Algorithm 1 Multi-agent flow matching with SE/PE guidance 1: Condition χ, decoder ND_N, and either a pretrained fiθf_i^θ or training data with encoder ℰNE_N. 2: if fiθf_i^θ is unavailable then 3: θ←argminθℒMAC(θ|χ) θ← _θL_MAC(θ\,|\,χ) using (5) 4: end if 5: (0)∼p0z(0) p_0. 6: if SE guidance then 7: Specify qaSEq_a^SE, βaSE _a^SE, αaSE _a^SE, and wa,iSEw_a,i^SE, and construct VaSEV_a^SE and ζa,iSE _a,i^SE using (12)–(13). 8: for all i∈i do 9: ui∗(τ)←u_i^*(τ)← solution of (15). 10: end for 11: else if PE guidance then 12: Choose αPEα^PE and ωPEω^PE according to Theorem 3.3. 13: for all i∈i do 14: ui∗(τ)←u_i^*(τ)← solution of (17). 15: end for 16: end if 17: Evolve (τ)z(τ) over τ∈[0,1]τ∈[0,1] according to (6) with ui=ui∗u_i=u_i^*. 18: return N((1)|χ)D_N(z(1)\,|\,χ). Figure 1: Snapshots of an N=5N=5 complete process of Task 1 with SE guidance. Top: the initial configuration. Middle: successful bridge construction. Bottom: spatial gap crossing. Figure 2: Evolution of qtaskq_ task and its time-varying upper bound βtask _ task during generative processes for policies generated at early (left), middle (center), and late (right) physical times in the manipulation stage of the example shown in Figure 1. 4 Applications In this section, we apply DeGG-Flow to two tasks with shared and private coupled requirements, respectively. Both applications are visualized in MuJoCo (Todorov et al., 2012). 4.1 Task 1: Multi-Robot Policy Generation Using SE Guidance Motivated by disaster response and search-and-rescue tasks, we study how multiple robots can collaborate and exploit tools in the environment to accomplish tasks beyond the capability of an individual robot. In Task 1, a team of robots faces a spatial gap that cannot be crossed directly, and collaboratively uses a plank to construct a traversable bridge. Since all robots act on the same rigid body, their generated motions must remain mutually compatible while satisfying shared task and safety requirements. Therefore, SE guidance is suitable for Task 1. The generated object is a motion policy specifying the robot contact point velocities on the plank. The condition χ contains the current robot and plank states and environment information. Only the first few actions of each generated policy are executed before a new policy is generated. Task 1 contains two SE factors: qtask(|χ)q_ task(z\,|\,χ), which encodes task progress at the plank pose reached after the executed actions, and qsafe(|χ)q_ safe(z\,|\,χ), which evaluates safety along these actions. The detailed settings are provided in Appendix B. For visualization, each complete process consists of three stages. The plank pose, robot initial and goal positions, and robot-plank contact points are randomly sampled subject to workspace constraints. First, the robots move to the contact points. Then, the learned multi-agent motion policy generates the collaborative motions for the robots to manipulate the plank to form a bridge over the spatial gap, with or without SE guidance. The success rates in Table 1 evaluate only this manipulation stage. After the bridge is successfully constructed, the robots move to the goals through the bridge. For evaluation, every robot uses the same trained GNN representing the nominal vector field fiθf_i^θ in (6). We refer to the generative process with ui=0u_i=0 as Nominal, and to the same nominal vector field with SE guidance giui∗g_iu_i^*, where ui∗u_i^* is obtained from (15), as Guided. Nominal and Guided use the same initial physical condition, initial sample (0)z(0), conditioning information χ, and decoder ND_N. The nominal vector field is trained with N∈4,5,6N∈\4,5,6\, and we evaluate 50 manipulation trials for each N∈4,5,6,7,8N∈\4,5,6,7,8\, with N=7,8N=7,8 evaluating generalization to team sizes unseen in the training data. Table 1: Success rates for Task 1 N Nominal Guided 44 46/5046/50 (92%) 50/5050/50 (100%) 55 45/5045/50 (90%) 50/5050/50 (100%) 66 43/5043/50 (86%) 50/5050/50 (100%) 77 (unseen) 42/5042/50 (84%) 50/5050/50 (100%) 88 (unseen) 42/5042/50 (84%) 50/5050/50 (100%) Overall 218/250218/250 (87.2%) 250/250250/250 (100%) As shown in Table 1, Nominal fails in 32 trials, whereas Guided successfully constructs a traversable bridge in all 250 trials, including the 100 trials with unseen team sizes of N=7,8N=7,8. Figure 1 visualizes an N=5N=5 Guided example. These results show that SE guidance can recover failures of the nominal generative policy while preserving successful collaboration at team sizes beyond those used during training. Additional results, including an N=7N=7 rescue example, and multi-seed training and validation curves, are provided in Appendix B.3. Figure 2 shows qtask((τ)|χ)q_ task(z(τ)\,|\,χ) and βtask(τ) _ task(τ) along τ in the generative process for policies generated at early, middle, and late physical times in the manipulation stage of the N=5N=5 Guided example shown in Figure 1. For visualization only, both curves in each plot are shifted by the same constant βtask(1) _ task(1), and qtask−βtaskq_ task- _ task is unchanged. The shaded region corresponds to qtask((τ)|χ)≤βtask(τ)q_ task(z(τ)\,|\,χ)≤ _ task(τ). As shown in Figure 2, early in physical time, qtaskq_ task starts close to its upper bound βtask _ task and then moves farther below it during the generative process. This is because βtask,start _ task,start is initialized just above the task factor value of the initial sample, and βtask,end=qtaskcur−0.05 _ task,end=q_ task cur-0.05 requires the generated policy to reduce the task factor by at least 0.050.05 relative to its value at the current plank pose. The SE guidance therefore progressively separates qtaskq_ task from the contracting upper bound while improving the task accomplishment. At the middle physical time, qtaskq_ task stays below βtask _ task even though it increases during the generative process. This is because the SE guidance does not require qtaskq_ task to decrease monotonically. Instead, it only requires qtask≤βtaskq_ task≤ _ task, allowing the nominal vector field to dominate the generated policy. Across the three physical times, the observed relation qtask((τ)|χ)≤βtask(τ)q_ task(z(τ)\,|\,χ)≤ _ task(τ) is consistent with Theorem 3.1. 4.2 Task 2: Multi-Object Scene Generation Using PE Guidance Section 4.1 demonstrates SE guidance in multi-robot collaboration. We next consider PE guidance in multi-object scene generation. A useful scene must satisfy not only geometric validity, such as collision avoidance and workspace containment, but also how individual objects are intended to be accessed or used. Each object therefore has its own affordance requirement, while whether that requirement is satisfied can depend on neighboring objects. Thus, PE guidance is suitable for Task 2. Task 2 generates a tabletop arrangement of N objects. For object i∈i , denote its pose by gOi=(Ri,pi)∈(2)g_O_i=(R_i,p_i) (2), and let N(|χ)=gOii∈D_N(z\,|\,χ)=\g_O_i\_i . The condition χ specifies the selected objects from a library, their geometry, the tabletop, and their affordance requirements. For each object, one or more nearby regions are specified and should be unobstructed when the object is used. We refer to these as affordance regions. For object i, let hik(|χ)h_ik(z\,|\,χ) denote its kkth geometric requirement, where hik≥0h_ik≥ 0 means that the requirement is satisfied. These requirements include object separation, unobstructed affordance regions, and containment of both objects and affordance regions within the tabletop. Let rik=−hikr_ik=-h_ik and choose a buffer δPE>0δ^PE>0. We define VPE=∑i=1NViPEV^PE= _i=1^NV_i^PE with ViPE(|χ)=∑k12maxrik(|χ)+δPE,02V_i^PE(z\,|\,χ)= _k 12 \! \r_ik(z\,|\,χ)+δ^PE,0 \^2. Detailed settings are provided in Appendix C. For evaluation, the nominal vector field is trained on scenes with N∈4,5,6N∈\4,5,6\, and we evaluate 50 cases for each N∈4,5,6,7,8N∈\4,5,6,7,8\. Natural Nominal evaluates scenes generated by the nominal vector field under the default affordance requirements. Private Nominal evaluates the same nominal scenes after the affordance requirements are changed, while Guided starts from the same initial samples and incorporates PE guidance based on the changed requirements. As shown in Table 2, the nominal success rate decreases as the number of objects increases, and Private Nominal further shows that a scene valid under the default requirements can become invalid when the desired access or use of an object changes. In contrast, Guided satisfies all requirements in all 250 cases, including the 100 cases with team sizes N=7,8N=7,8 unseen in the training data. Figure 3 shows an N=7N=7 example in which PE guidance adapts a nominal scene from a right-handed to a left-handed laptop use requirement without retraining the generative model. These results demonstrate that PE guidance can result in generated objects satisfying private requirements, including at team sizes beyond those used during training. Additional results, including an N=8N=8 example with multiple simultaneous violations resulting from the nominal vector field, results across different affordance requirements, and multi-seed training and validation curves are provided in Appendix C.2. Table 2: Success rates for Task 2 N Natural Nominal Private Nominal Guided 44 44/5044/50 (88%) 36/5036/50 (72%) 50/5050/50 (100%) 55 35/5035/50 (70%) 31/5031/50 (62%) 50/5050/50 (100%) 66 32/5032/50 (64%) 26/5026/50 (52%) 50/5050/50 (100%) 77 (unseen) 16/5016/50 (32%) 13/5013/50 (26%) 50/5050/50 (100%) 88 (unseen) 8/508/50 (16%) 6/506/50 (12%) 50/5050/50 (100%) Overall 135/250135/250 (54.0%) 112/250112/250 (44.8%) 250/250250/250 (100%) Figure 3: An N=7N=7 example for Task 2. Green circles represent the objects’ affordance regions that are unobstructed, while the red circle represents one that is obstructed. Left: the scene generated using the nominal vector field satisfies the default right-handed laptop use requirement. Center: the same scene evaluated for a left-handed user, where the required mouse operating region on the left side of the laptop is occupied by a potted plant. Right: PE guidance generates a scene satisfying the left-handed use requirement. Figure 4 shows the evolution of VPEV^PE during the generative process for the N=7N=7 example corresponding to Figure 3. Both trajectories start from the same initial Gaussian sample and are evaluated under the same left-handed laptop use requirement, which requires the mouse operating region on the left side of the laptop to be unobstructed. Without guidance, this requirement is violated at the end of the generative process, whereas PE guidance guarantees that VPEV^PE converges to zero and generates a scene satisfying the new requirement without retraining, which is consistent with Theorem 3.3. Figure 4: Evolution of VPEV^PE during the generative process for the N=7N=7 example in Figure 3. 5 Conclusion This paper proposes DeGG-Flow, a general framework for addressing coupled multi-agent requirements through decoupled generative guidance, without applying a separate correction stage after generation. Each agent determines its own guidance without relying on the simultaneously computed guidance of any other agents, so the team need not rely on a central decision maker. We establish feasibility and finite-horizon guarantees for both SE and PE guidance, and demonstrate the effectiveness of DeGG-Flow in multi-robot collaboration and multi-object scene generation, including numbers of agents beyond those used in training. More broadly, DeGG-Flow allows a pretrained generative model to accommodate new hard requirements or constraints that are unseen in its training data, without the need of collecting new data or retraining the model. Future work will investigate discrete-time theoretical guarantees and large-scale multi-agent systems in real world. References Albergo et al. (2025) M. Albergo, N. M. Boffi, and E. Vanden-Eijnden Stochastic interpolants: a unifying framework for flows and diffusions. Journal of Machine Learning Research 26 (209), p. 1–80. Cited by: §1. Ames et al. (2016) A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada Control barrier function based quadratic programs for safety critical systems. IEEE transactions on automatic control 62 (8), p. 3861–3876. Cited by: §1. Bansal et al. (2024) A. Bansal, H. Chu, A. Schwarzschild, R. Sengupta, M. Goldblum, J. Geiping, and T. Goldstein Universal guidance for diffusion models. In International Conference on Learning Representations, Vol. 2024, p. 51304–51323. Cited by: §1. Bhat and Bernstein (2000) S. P. Bhat and D. S. Bernstein Finite-time stability of continuous autonomous systems. SIAM Journal on Control and optimization 38 (3), p. 751–766. Cited by: §A.3. Brown et al. (2020) T. Brown, B. Mann, N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, et al. Language models are few-shot learners. Advances in neural information processing systems 33, p. 1877–1901. Cited by: §1. Chi et al. (2025) C. Chi, Z. Xu, S. Feng, E. Cousineau, Y. Du, B. Burchfiel, R. Tedrake, and S. Song Diffusion policy: visuomotor policy learning via action diffusion. The International Journal of Robotics Research 44 (10-11), p. 1684–1704. Cited by: §1. Dai et al. (2025) X. Dai, Z. Yang, D. Yu, F. Liu, H. Sadeghian, S. Haddadin, and S. Hirche SafeFlow: safe robot motion planning with flow matching via control barrier functions. arXiv preprint arXiv:2504.08661. Cited by: §1. Feng et al. (2025) R. Feng, C. Yu, W. Deng, P. Hu, and T. Wu On the guidance of flow matching. arXiv preprint arXiv:2502.02150. Cited by: §1. Gadginmath et al. (2026) D. Gadginmath, A. Allibhoy, and F. Pasqualetti Provably safe generative sampling with constricting barrier functions. arXiv preprint arXiv:2602.21429. Cited by: §1. Gilmer et al. (2017) J. Gilmer, S. S. Schoenholz, P. F. Riley, O. Vinyals, and G. E. Dahl Neural message passing for quantum chemistry. In International conference on machine learning, p. 1263–1272. Cited by: §2.1. Ho et al. (2020) J. Ho, A. Jain, and P. Abbeel Denoising diffusion probabilistic models. Advances in neural information processing systems 33, p. 6840–6851. Cited by: §1, §1. Janner et al. (2022) M. Janner, Y. Du, J. B. Tenenbaum, and S. Levine Planning with diffusion for flexible behavior synthesis. arXiv preprint arXiv:2205.09991. Cited by: §1. Kschischang et al. (2001) F. R. Kschischang, B. J. Frey, and H. Loeliger Factor graphs and the sum-product algorithm. IEEE Transactions on information theory 47 (2), p. 498–519. Cited by: §2.2. Lee et al. (2026) D. Lee, D. Lee, and A. Zhang Multi-agent coordination via flow matching. In International Conference on Learning Representations, Vol. 2026, p. 99912–99943. Cited by: §1. Liang et al. (2025) J. Liang, J. K. Christopher, S. Koenig, and F. Fioretto Simultaneous multi-robot motion planning with projected diffusion models. arXiv preprint arXiv:2502.03607. Cited by: §1. Lin et al. (2026) R. Lin, G. Notomista, and M. Egerstedt Disentangled control of multi-agent systems. IEEE Transactions on Control of Network Systems. Cited by: 1st item, §2.2. Lipman et al. (2022) Y. Lipman, R. T. Chen, H. Ben-Hamu, M. Nickel, and M. Le Flow matching for generative modeling. arXiv preprint arXiv:2210.02747. Cited by: §1, §2.1. Parimi and Williams (2025) V. Parimi and B. C. Williams Diffusion-guided multi-arm motion planning. arXiv preprint arXiv:2509.08160. Cited by: §1. Park (1995) F. C. Park Distance metrics on the rigid-body motions with applications to mechanism design. Journal of Mechanical Design. Cited by: §C.2. Rombach et al. (2022) R. Rombach, A. Blattmann, D. Lorenz, P. Esser, and B. Ommer High-resolution image synthesis with latent diffusion models. In 2022 IEEE/CVF conference on computer vision and pattern recognition (CVPR), p. 10674–10685. Cited by: §1. Shaoul et al. (2025) Y. Shaoul, I. Mishani, S. Vats, J. Li, and M. Likhachev Multi-robot motion planning with diffusion models. In International Conference on Learning Representations, Vol. 2025, p. 95791–95811. Cited by: §1. Tang et al. (2024) J. Tang, Y. Nie, L. Markhasin, A. Dai, J. Thies, and M. Nießner Diffuscene: denoising diffusion models for generative indoor scene synthesis. In 2024 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), p. 20507–20518. Cited by: §1. Todorov et al. (2012) E. Todorov, T. Erez, and Y. Tassa Mujoco: a physics engine for model-based control. In 2012 IEEE/RSJ international conference on intelligent robots and systems, p. 5026–5033. Cited by: §4. Xiao et al. (2025) W. Xiao, J. Wang, C. Gan, R. Hasani, M. Lechner, and D. Rus Safediffuser: safe planning with diffusion probabilistic models. In International Conference on Learning Representations, Vol. 2025, p. 100391–100412. Cited by: §1. Yang et al. (2026) J. Yang, S. Jang, and S. Han Safeflowmatcher: safe and fast planning using flow matching with control barrier functions. In International Conference on Learning Representations, Vol. 2026, p. 54058–54087. Cited by: §1. Zhu et al. (2024) Z. Zhu, M. Liu, L. Mao, B. Kang, M. Xu, Y. Yu, S. Ermon, and W. Zhang Madiff: offline multi-agent learning with diffusion models. Advances in Neural Information Processing Systems 37, p. 4177–4206. Cited by: §1. Appendix A Proofs A.1 Proof of Theorem 3.1 For any i∈aSEi _a^SE, (12) gives ∇ziVaSE(τ,aSE|χ)=∇ziqaSE(τ,aSE|χ). _z_iV_a^SE(τ,z_S_a^SE\,|\,χ)= _z_iq_a^SE(τ,z_S_a^SE\,|\,χ). Along a trajectory of (6), dτVaSE(τ,aSE(τ)|χ) ddτV_a^SE(τ,z_S_a^SE(τ)\,|\,χ) (21) = = ∂τqaSE(τ,aSE(τ)|χ)−β˙aSE(τ|χ,(0)) _τq_a^SE(τ,z_S_a^SE(τ)\,|\,χ)- β_a^SE(τ\,|\,χ,z(0)) +∑i∈aSE∇ziqaSE(τ,aSE(τ)|χ)⊤ + _i _a^SE _z_iq_a^SE(τ,z_S_a^SE(τ)\,|\,χ) ⋅(fiθ(τ,(τ)|χ)+gi(τ,(τ)|χ)ui(τ)). · (f_i^θ(τ,z(τ)\,|\,χ)+g_i(τ,z(τ)\,|\,χ)u_i(τ) ). According to (13), (7) becomes ∇ziqaSE(τ,aSE(τ)|χ)⊤ _z_iq_a^SE(τ,z_S_a^SE(τ)\,|\,χ) (22) ⋅(fiθ(τ,(τ)|χ)+gi(τ,(τ)|χ)ui(τ)) · (f_i^θ(τ,z(τ)\,|\,χ)+g_i(τ,z(τ)\,|\,χ)u_i(τ) ) + + wa,iSE(∂τqaSE(τ,aSE(τ)|χ)−β˙aSE(τ|χ,(0))CLOSE w_a,i^SE ( _τq_a^SE(τ,z_S_a^SE(τ)\,|\,χ)- β_a^SE(τ\,|\,χ,z(0)) OPEN+αaSE(VaSE(τ,aSE(τ)|χ)))≤0. + _a^SE\! (V_a^SE(τ,z_S_a^SE(τ)\,|\,χ) ) )≤ 0. Summing (22) over i∈aSEi _a^SE and using wa,iSE≥0w_a,i^SE≥ 0 such that ∑i∈aSEwa,iSE=1 _i _a^SEw_a,i^SE=1 yields ∑i∈aSE∇ziqaSE(τ,aSE(τ)|χ)⊤ _i _a^SE _z_iq_a^SE(τ,z_S_a^SE(τ)\,|\,χ) (23) ⋅(fiθ(τ,(τ)|χ)+gi(τ,(τ)|χ)ui(τ)) · (f_i^θ(τ,z(τ)\,|\,χ)+g_i(τ,z(τ)\,|\,χ)u_i(τ) ) +∂τqaSE(τ,aSE(τ)|χ)−β˙aSE(τ|χ,(0)) + _τq_a^SE(τ,z_S_a^SE(τ)\,|\,χ)- β_a^SE(τ\,|\,χ,z(0)) +αaSE(VaSE(τ,aSE(τ)|χ))≤0. + _a^SE\! (V_a^SE(τ,z_S_a^SE(τ)\,|\,χ) )≤ 0. According to (23) and (21), we have dτVaSE(τ,aSE(τ)|χ)+αaSE(VaSE(τ,aSE(τ)|χ))≤0. ddτV_a^SE(τ,z_S_a^SE(τ)\,|\,χ)+ _a^SE\! (V_a^SE(τ,z_S_a^SE(τ)\,|\,χ) )≤ 0. (24) Since αaSE _a^SE is an extended class-∞K_∞ function, we have αaSE(0)=0 _a^SE(0)=0 and αaSE(s)∈ℝ≥0 _a^SE(s) _≥ 0, ∀s∈ℝ≥0∀ s _≥ 0. Suppose VaSE(0,aSE(0)|χ)≤0V_a^SE(0,z_S_a^SE(0)\,|\,χ)≤ 0 but there exists some τ1∈(0,1] _1∈(0,1] such that VaSE(τ1,aSE(τ1)|χ)∈ℝ>0V_a^SE( _1,z_S_a^SE( _1)\,|\,χ) _>0. By continuity, let τ0<τ1 _0< _1 be the last time before τ1 _1 at which VaSE(τ0,aSE(τ0)|χ)=0V_a^SE( _0,z_S_a^SE( _0)\,|\,χ)=0. Then, for every τ∈(τ0,τ1]τ∈( _0, _1], VaSE(τ,aSE(τ)|χ)∈ℝ>0V_a^SE(τ,z_S_a^SE(τ)\,|\,χ) _>0, αaSE(VaSE(τ,aSE(τ)|χ))∈ℝ>0 _a^SE\! (V_a^SE(τ,z_S_a^SE(τ)\,|\,χ) ) _>0, and dτVaSE(τ,aSE(τ)|χ)<0 ddτV_a^SE(τ,z_S_a^SE(τ)\,|\,χ)<0 by (24), which contradicts VaSE(τ0,aSE(τ0)|χ)=0V_a^SE( _0,z_S_a^SE( _0)\,|\,χ)=0 and VaSE(τ1,aSE(τ1)|χ)∈ℝ≥0V_a^SE( _1,z_S_a^SE( _1)\,|\,χ) _≥ 0. Therefore, VaSE(τ,aSE(τ)|χ)≤0,∀τ∈[0,1],V_a^SE(τ,z_S_a^SE(τ)\,|\,χ)≤ 0, ∀τ∈[0,1], and (12) at τ=1τ=1 gives qaSE(1,aSE(1)|χ)≤βaSE(1|χ,(0))q_a^SE(1,z_S_a^SE(1)\,|\,χ)≤ _a^SE(1\,|\,χ,z(0)). A.2 Proof of Proposition 3.2 In this paper, we choose ϕδ(η)=η,η≤−δ,δ(η/δ−1)3(η/δ+3)/16,|η|<δ,0,η≥δ, _δ(η)= casesη,&η≤-δ,\\ δ (η/δ-1 )^3 (η/δ+3 )/16,&|η|<δ,\\ 0,&η≥δ, cases which satisfies that ϕδ∈C2 _δ∈ C^2, ϕδ(η)∈[−1,0] _δ(η)∈[-1,0], and ϕδ(η)≤minη,0 _δ(η)≤ \η,0\, ∀η∈[−1,1]∀η∈[-1,1]. Given an agnet i, assume it is incident to two active SE factors, indexed by 11 and 22, at any τ∈[0,1]τ∈[0,1]. Since ϕδ(η)≤η _δ(η)≤η, then we have (ℓ2,iSE)⊤d1,iSE ( _2,i^SE) d_1,i^SE =‖ℓ1,iSE‖‖ℓ2,iSE‖(ηiSE−ϕδ(ηiSE))≥0, = \| _1,i^SE \| \| _2,i^SE \| ( _i^SE- _δ( _i^SE) )≥ 0, (ℓ1,iSE)⊤d2,iSE ( _1,i^SE) d_2,i^SE =‖ℓ1,iSE‖‖ℓ2,iSE‖(ηiSE−ϕδ(ηiSE))≥0, = \| _1,i^SE \| \| _2,i^SE \| ( _i^SE- _δ( _i^SE) )≥ 0, (ℓ1,iSE)⊤d1,iSE ( _1,i^SE) d_1,i^SE =‖ℓ1,iSE‖2(1−ηiSEϕδ(ηiSE))≥0, = \| _1,i^SE \|^2 (1- _i^SE _δ( _i^SE) )≥ 0, (ℓ2,iSE)⊤d2,iSE ( _2,i^SE) d_2,i^SE =‖ℓ2,iSE‖2(1−ηiSEϕδ(ηiSE))≥0, = \| _2,i^SE \|^2 (1- _i^SE _δ( _i^SE) )≥ 0, where the last two quantities are strictly positive whenever the corresponding da,iSE≠0d_a,i^SE≠ 0 since ηiSE∈[−1,1] _i^SE∈[-1,1] and ϕδ(ηiSE)∈[−1,0] _δ( _i^SE)∈[-1,0]. Additionally, for every active SE factor a, (14) satisfies wa,iSE≥0w_a,i^SE≥ 0 and ∑i∈aSEwa,iSE=1 _i _a^SEw_a,i^SE=1. Also, consider the SE inequalities (ℓa,iSE)⊤ui+ba,iSE≤0,∀a∈1,2.( _a,i^SE) u_i+b_a,i^SE≤ 0, ∀ a∈\1,2\. (25) Define λa,i≔ba,iSE/((ℓa,iSE)⊤da,iSE) _a,i b_a,i^SE/(( _a,i^SE) d_a,i^SE) when ba,iSE>0b_a,i^SE>0, and λa,i≔0 _a,i 0 otherwise. One can observe that (ℓa,iSE)⊤da,iSE>0( _a,i^SE) d_a,i^SE>0 whenever ba,iSE>0b_a,i^SE>0 since da,iSE≠0d_a,i^SE≠ 0 in this case. Now choose ui=−λ1,id1,iSE−λ2,id2,iSE.u_i=- _1,id_1,i^SE- _2,id_2,i^SE. Then, for a=1a=1, (ℓ1,iSE)⊤ui+b1,iSE=−λ1,i(ℓ1,iSE)⊤d1,iSE−λ2,i(ℓ1,iSE)⊤d2,iSE+b1,iSE.( _1,i^SE) u_i+b_1,i^SE=- _1,i( _1,i^SE) d_1,i^SE- _2,i( _1,i^SE) d_2,i^SE+b_1,i^SE. (26) If b1,iSE>0b_1,i^SE>0, the first and last terms on the right-hand side of (26) cancel, and the remaining term is nonpositive. If b1,iSE≤0b_1,i^SE≤ 0, then λ1,i=0 _1,i=0 and the left-hand side of (26) is nonpositive. The same analysis holds for a=2a=2. Hence, (25) are satisfied. If only one SE factor is active, then da,iSE=ℓa,iSEd_a,i^SE= _a,i^SE, and the same construction with a single coefficient λa,i _a,i also gives a feasible uiu_i. Therefore, the feasible set of (15) is nonempty. A.3 Proof of Theorem 3.3 Since ∂τViPE(τ,|χ)=0 _τV_i^PE(τ,z\,|\,χ)=0, ∀i∈∀ i , we have dτVPE((τ)|χ) ddτV^PE(z(τ)\,|\,χ) (27) = = ∑i∈∇ziVPE((τ)|χ)⊤ _i _z_iV^PE(z(τ)\,|\,χ) ⋅(fiθ(τ,(τ)|χ)+gi(τ,(τ)|χ)ui(τ)). · (f_i^θ(τ,z(τ)\,|\,χ)+g_i(τ,z(τ)\,|\,χ)u_i(τ) ). Using (16) in (10), summing (9) over i∈i , and applying (27) yield dτVPE((τ)|χ)+cPEωPE(τ)∑i∈(ViPE((τ)|χ))ρPE≤0. ddτV^PE(z(τ)\,|\,χ)+c^PEω^PE(τ) _i (V_i^PE(z(τ)\,|\,χ) )^ρ^PE≤ 0. Since ρPE∈(0,1)ρ^PE∈(0,1), αPE(s)=cPEsρPEα^PE(s)=c^PEs^ρ^PE is subadditive on ℝ≥0R_≥ 0. Hence, αPE(VPE((τ)|χ))≤∑i∈αPE(ViPE((τ)|χ)),α^PE\! (V^PE(z(τ)\,|\,χ) )≤ _i α^PE\! (V_i^PE(z(τ)\,|\,χ) ), and thus dτVPE((τ)|χ)≤−cPEωPE(τ)(VPE((τ)|χ))ρPE, ddτV^PE(z(τ)\,|\,χ)≤-c^PEω^PE(τ) (V^PE(z(τ)\,|\,χ) )^ρ^PE, (28) which adapts the finite-time Lyapunov inequality (Bhat and Bernstein, 2000) to the generative process using the time-varying weight ωPE(τ)ω^PE(τ). On any interval over which VPE((τ)|χ)>0V^PE(z(τ)\,|\,χ)>0, multiplying (28) by (1−ρPE)(VPE((τ)|χ))−ρPE(1-ρ^PE) (V^PE(z(τ)\,|\,χ) )^-ρ^PE leads to dτ(VPE((τ)|χ))1−ρPE≤−cPE(1−ρPE)ωPE(τ). ddτ (V^PE(z(τ)\,|\,χ) )^1-ρ^PE≤-c^PE(1-ρ^PE)\,ω^PE(τ). Integrating from 00 to τ while VPE((τ)|χ)>0V^PE(z(τ)\,|\,χ)>0 results in (VPE((τ)|χ))1−ρPE (V^PE(z(τ)\,|\,χ) )^1-ρ^PE ≤ ≤ (VPE((0)|χ))1−ρPE−cPE(1−ρPE)∫0τωPE(s)ds. (V^PE(z(0)\,|\,χ) )^1-ρ^PE-c^PE(1-ρ^PE) _0^τ\!ω^PE(s)\,ds. Additionally, (28) and VPE≥0V^PE≥ 0 imply that, for any τ¯∈[0,1] τ∈[0,1], VPE((τ¯)|χ)=0⟹VPE((τ)|χ)=0,∀τ∈[τ¯,1].V^PE(z( τ)\,|\,χ)=0\;\; \;\;V^PE(z(τ)\,|\,χ)=0,\;∀τ∈[ τ,1]. Therefore, VPE((τ)|χ) V^PE(z(τ)\,|\,χ) ≤max0,(VPE((0)|χ))1−ρPE ≤ \0,\; (V^PE(z(0)\,|\,χ) )^1-ρ^PE −cPE(1−ρPE)∫0τωPE(s)ds11−ρPE, -c^PE(1-ρ^PE) _0^τ\!ω^PE(s)\,ds \ 11-ρ^PE, and evaluating the upper bound at τ=1τ=1 implies VPE((1)|χ)=0V^PE(z(1)\,|\,χ)=0. A.4 Proof of Corollary 3.4 If biPE≤0b_i^PE≤ 0, then ui=0u_i=0 satisfies the inequality constraint in (17). If biPE>0b_i^PE>0 and ℓiPE≠0 _i^PE≠ 0, choose ui=−biPE(ℓiPE)⊤Hi−1ℓiPEHi−1ℓiPE.u_i=- b_i^PE( _i^PE) H_i^-1 _i^PEH_i^-1 _i^PE. Since Hi≻0H_i 0 and ℓiPE≠0 _i^PE≠ 0, we have (ℓiPE)⊤Hi−1ℓiPE>0( _i^PE) H_i^-1 _i^PE>0, and thus (ℓiPE)⊤ui+biPE=0( _i^PE) u_i+b_i^PE=0. Therefore, the feasible set of (17) is nonempty. A.5 Proof of Theorem 3.5 For an initial sample ξ∼p0ξ p_0, define Δ(τ|ξ)≔gui(τ|ξ)−nom(τ|ξ). (τ\,|\,ξ) ^gui(τ\,|\,ξ)-X^nom(τ\,|\,ξ). Since the nominal and guided trajectories start from the same initial sample, Δ(0|ξ)=0 (0\,|\,ξ)=0. Then, we can obtain Δ(τ|ξ)=∫0τ(CLOSE (τ\,|\,ξ)= _0^τ\! ( fθ(s,gui(s|ξ)|χ) f^θ(s,X^gui(s\,|\,ξ)\,|\,χ) OPEN−fθ(s,nom(s|ξ)|χ)+Γ(s|ξ))ds. -f^θ(s,X^nom(s\,|\,ξ)\,|\,χ)+ (s\,|\,ξ) )\,ds. Since ‖fθ(τ,|χ)−fθ(τ,|χ)‖≤L(τ)‖−‖ \|f^θ(τ,y\,|\,χ)-f^θ(τ,z\,|\,χ) \|≤ L(τ) \|y-z \|, ∀τ∈[0,1]∀τ∈[0,1], ∀,∈N ,z ^N, we have ‖Δ(τ|ξ)‖≤∫0τL(s)‖Δ(s|ξ)‖s+∫0τ‖Γ(s|ξ)‖s. \| (τ\,|\,ξ) \|≤ _0^τ\!L(s) \| (s\,|\,ξ) \|\,ds+ _0^τ\! \| (s\,|\,ξ) \|\,ds. Then, applying Grönwall’s inequality leads to ‖Δ(1|ξ)‖≤∫01exp(∫s1L(r)r)‖Γ(s|ξ)‖s. \| (1\,|\,ξ) \|≤ _0^1 \! ( _s^1L(r)\,dr ) \| (s\,|\,ξ) \|\,ds. Thus, using Minkowski’s inequality, we can obtain ∫N‖Δ(1|ξ)‖2p0(ξ)ξ _Z^N\! \| (1\,|\,ξ) \|^2p_0(ξ)\,dξ (29) ≤ ≤ ∫01exp(∫s1L(r)dr)∫N‖Γ(s|ξ)‖2p0(ξ)ξds. _0^1\! \! ( _s^1\!L(r)\,dr ) _Z^N\! \| (s\,|\,ξ) \|^2p_0(ξ)\,dξ\;ds. Since the Wasserstein distance between μ and ν with finite second moments is W22(μ,ν)≔infπ∈Π(μ,ν)∫‖−‖2π(,),W_2^2(μ,ν) _π∈ (μ,ν) \! \|x-y \|^2\,dπ(x,y), (30) where Π(μ,ν) (μ,ν) denotes the set of probability measures π on N×NZ^N×Z^N whose first and second marginal distributions are μ and ν, respectively, we have W22(μ1,ν1)≤ W_2^2( _1, _1)≤ ∫N‖gui(1|ξ)−nom(1|ξ)‖2p0(ξ)ξ _Z^N\! \|X^gui(1\,|\,ξ)-X^nom(1\,|\,ξ) \|^2p_0(ξ)\,dξ =∫N‖Δ(1|ξ)‖2p0(ξ)dξ. = _Z^N\! \| (1\,|\,ξ) \|^2p_0(ξ)\,dξ. Using(29), (19) can be obtained. In addition, for a π∈Π(μ1,ν1)π∈ ( _1, _1), let (nom,gui)(Z^nom,Z^gui) have joint probability distribution π. Then N(nom|χ)D_N(Z^nom\,|\,χ) and N(gui|χ)D_N(Z^gui\,|\,χ) have probability distributions μ¯1 μ_1 and ν¯1 ν_1, respectively. Thus, according to (30) and ‖N(|χ)−N(|χ)‖≤LD‖−‖ \|D_N(y\,|\,χ)-D_N(z\,|\,χ) \|≤ L_D \|y-z \|, ∀,∈N ,z ^N, we can obtain W22(μ¯1,ν¯1) W_2^2( μ_1, ν_1) ≤∫‖N(|χ)−N(|χ)‖2dπ(,) ≤ \! \|D_N(x\,|\,χ)-D_N(y\,|\,χ) \|^2\,dπ(x,y) ≤LD2∫‖−‖2dπ(,). ≤ L_D^2 \! \|x-y \|^2\,dπ(x,y). Since π∈Π(μ1,ν1)π∈ ( _1, _1) is arbitrary, we have W22(μ¯1,ν¯1)≤LD2W22(μ1,ν1),W_2^2( μ_1, ν_1)≤ L_D^2W_2^2( _1, _1), which gives (20). Appendix B Task 1: Detailed Settings and Additional Results B.1 Representation, conditioning, network structure, and training Denote the plank pose as gB∈(2)g_B (2), with orientation RB∈(2)R_B (2) and position pB∈ℝ2p_B ^2. Its body twist is ξBb=(gB−1g˙B)∨=[(vBb)⊤,ωBb]⊤∈ℝ3, _B^b=(g_B^-1 g_B) =[(v_B^b) , _B^b] ^3, where (⋅)∨:(2)→ℝ3(·) : se(2) ^3. For robot i, let ri=[ri,x,ri,y]⊤∈ℝ2r_i=[r_i,x,r_i,y] ^2 denote its contact point in the body frame of the plank and define ri⟂≔[−ri,y,ri,x]⊤r_i [-r_i,y,\,r_i,x] . The velocity of the contact point is cib=JB(ri)ξBb,c_i^b=J_B(r_i) _B^b, where JB(ri)≔[I2,ri⟂]∈ℝ2×3J_B(r_i) [I_2,r_i ] ^2× 3. The object generated in Task 1 is a motion policy over a finite physical horizon. Conditioned on χ, the decoder maps the joint generative state to a sequence of robot contact-point velocities, N(|χ)=ci,tbi∈,t∈,D_N(z\,|\,χ)=\c_i,t^b\_i ,\,t , where ≔1,…,TT \1,…,T\ and ci,tb=JB(ri)ξB,tb(|χ)c_i,t^b=J_B(r_i) _B,t^b(z\,|\,χ). Only the first TexecT_ exec actions of the generated policy are executed before a new policy is generated. Let ≔1,…,KK \1,…,K\. For each robot, the generative state zi∈ℝ24z_i ^24 is decoded into K=8K=8 three-dimensional vectors ηi,k∈ℝ3 _i,k ^3, ∀k∈∀ k . Define Mi≔JB(ri)⊤JB(ri)M_i J_B(r_i) J_B(r_i) and G≔∑i=1NMiG _i=1^NM_i, with G≻0G 0. Let ≔1,…,TT \1,…,T\ with T=20T=20, and let Bspl∈ℝT×KB_ spl ^T× K be the B-spline matrix. Before a new policy is generated, a T-step body twist sequence ξ¯B,tbt∈\ ξ_B,t^b\_t is constructed with the previously generated policy and the current motion. Then, the decoder N(|χ)=ci,tbi∈,t∈D_N(z\,|\,χ)=\c_i,t^b\_i ,t gives ξB,tb=ξ¯B,tb+∑k=1K(Bspl)t,k(G−1∑i=1NMiηi,k), _B,t^b= ξ_B,t^b+ _k=1^K(B_ spl)_t,k(G^-1 _i=1^NM_i _i,k), and ci,tb=JB(ri)ξB,tbc_i,t^b=J_B(r_i) _B,t^b, ∀i∈,t∈∀ i ,t . Only the first Texec=4T_ exec=4 actions of the generated T-step policy are executed before a new policy is generated. Before the next initial sample is drawn from a Gaussian distribution, the previous policy is shifted forward by these four actions, with its final value repeated to keep length T, and its first five values interpolated between the current motion and the shifted sequence. The resulting sequence ξ¯B,tbt∈\ ξ_B,t^b\_t is included in the next condition χ. For Task 1, the condition χ contains only quantities known before a new policy is generated and is characterized by three vectors for each robot i. Specifically, the vector λi∈ℝ23 _i ^23 represents robot i, the plank, and their locations relative to the workspace geometry. For each other robot j≠ij≠ i, the vector πij∈ℝ8 _ij ^8 represents the position and velocity of robot j relative to robot i. The vector ci∈ℝ45c_i ^45 represents robot i’s contact location on the plank, the desired plank pose, and the continuation of its contact point velocities from the previously generated policy. In particular, let piw,viw∈ℝ2p_i^w,v_i^w ^2 be the current position and velocity of robot i in the world frame. Denote the current plank pose as gB=(RB,pB)g_B=(R_B,p_B) with pB=[xB,yB]⊤p_B=[x_B,y_B] , and let θB _B denote the angle associated with RBR_B. Let vBw∈ℝ2v_B^w ^2 and ωBb∈ℝ _B^b be the current translational and angular velocities of the plank. The corresponding quantities in the body frame of the plank are Δpib≔RB⊤(piw−pB) p_i^b R_B (p_i^w-p_B), vib≔RB⊤viwv_i^b R_B v_i^w, and vBb≔RB⊤vBwv_B^b R_B v_B^w. Let [Xmin,Xmax]×[Ymin,Ymax][X_ ,X_ ]×[Y_ ,Y_ ] quantify the workspace, define X≔(Xmax−Xmin)/2X (X_ -X_ )/2 and Y≔(Ymax−Ymin)/2Y (Y_ -Y_ )/2, and let xgapLx_ gap^L and xgapRx_ gap^R be the two x-coordinates of the gap edges. We denote by xgoalminx_ goal the minimum x-coordinate of the region from which robot goals are sampled. The vector eB∈ℝ11e_B ^11 consists of the plank orientation and its position relative to the workspace, i.e., eB:=[ e_B:= [ cos(θB),−sin(θB),sin(θB),cos(θB), ( _B),\,- ( _B),\, ( _B),\, ( _B), Xmin−xBX,xgapL−xBX,xgapR−xBX,xgoalmin−xBX, X_ -x_BX,\, x_ gap^L-x_BX,\, x_ gap^R-x_BX,\, x_ goal -x_BX, Xmax−xBX,Ymin−yBY,Ymax−yBY]⊤. X_ -x_BX,\, Y_ -y_BY,\, Y_ -y_BY ] . Denote oi∈0,13o_i∈\0,1\^3, which indicates if robot i is on the initial side ground, on the plank, or on the opposite side ground. Let ai∈0,1a_i∈\0,1\ indicate whether robot i contacts with the plank, and let bB∈0,1b_B∈\0,1\ indicate whether bridge construction has succeeded. Additionally, λi=[ _i= [ (diag(1X,1Y)Δpib)⊤, (diag\! ( 1X, 1Y ) p_i^b ) , (vib0.86)⊤,oi⊤,(vBb0.44)⊤, ( v_i^b0.86 ) ,o_i , ( v_B^b0.44 ) , ωBb0.32,bB,ai,eB⊤]⊤∈ℝ23. _B^b0.32,b_B,a_i,e_B ] ^23. For every j≠ij≠ i, denote the relative position and velocity Δpijb≔RB⊤(pjw−piw) p_ij^b R_B (p_j^w-p_i^w) and Δvijb≔RB⊤(vjw−viw) v_ij^b R_B (v_j^w-v_i^w) in the body frame of the plank, then πij=[ _ij= [ (diag(1X,1Y)Δpijb)⊤, (diag\! ( 1X, 1Y ) p_ij^b ) , (Δvijb0.86)⊤,oj⊤,aj]⊤∈ℝ8. ( v_ij^b0.86 ) ,o_j ,a_j ] ^8. Let LBL_B be the plank length, and the desired plank pose be gBgoal=(Rg(θg),pg)g_B goal=(R_g( _g),p_g). Denote RB⊤(pg−pB)=[Δxgb,Δygb]⊤R_B (p_g-p_B)=[ x_g^b, y_g^b] , and let Δθg∈(−π,π] _g∈(-π,π] be the shortest signed angular difference from θB _B to θg _g. Let Cicont=[JB(ri)ξ¯B,1b,…,JB(ri)ξ¯B,Tb]⊤∈ℝT×2.C_i cont= [J_B(r_i) ξ_B,1^b,…,J_B(r_i) ξ_B,T^b ] ^T× 2. For each of the 4040 entries of CicontC_i cont, the corresponding training set mean is subtracted and the result is divided by the corresponding training set standard deviation. Let C~icont∈ℝT×2 C_i cont ^T× 2 denote the resulting matrix, then ci=[ c_i= [ ri,xLB/2,sgn(ri,y),ΔxgbLB, r_i,xL_B/2,sgn(r_i,y), x_g^bL_B, ΔygbY,Δθgπ/3,vec(C~icont)⊤]⊤∈ℝ45, y_g^bY, _gπ/3,vec( C_i cont) ] ^45, where sgn(s)=1sgn(s)=1 for s>0s>0 and sgn(s)=−1sgn(s)=-1 for s<0s<0, and vec(⋅)vec(·) stacks the entries of a matrix into a vector. For each j≠ij≠ i, the multilayer perceptron (MLP) ψmsg _ msg, with layer dimensions 77→160→16077→ 160→ 160, is applied to zjz_j, cjc_j, and πij _ij to form the message vector mij=ψmsg([zj⊤,cj⊤,πij⊤]⊤)∈ℝ160.m_ij= _ msg\! ([z_j ,c_j , _ij ] ) ^160. The message vectors from all other robots are averaged as m¯i≔1N−1∑j≠imij. m_i 1N-1 _j≠ im_ij. The MLP ψag _ ag shared by each agent, with layer dimensions 258→256→256→256→24258→ 256→ 256→ 256→ 24, is used to represent fiθ(τ,|χ)=ψag(CLOSE f_i^θ(τ,z\,|\,χ)= _ ag\! ( [zi⊤,ci⊤,λi⊤,m¯i⊤, [z_i ,c_i , _i , m_i , (N−1)/7,φ(τ)⊤]⊤)∈ℝ24, (N-1)/7, (τ) ] ) ^24, in which φ(τ)=[τ,sin(πτ),cos(πτ),sin(2πτ),cos(2πτ)]⊤∈ℝ5. (τ)=[τ, (πτ), (πτ), (2πτ), (2πτ)] ^5. The nominal vector field is trained and validated on N∈4,5,6N∈\4,5,6\. For Task 1, we instantiate the conditional probability path by choosing p0=(0,I)p_0=N(0,I). For an encoded data sample ∼pdata(|χ)Z p_ data(z\,|\,χ), we draw 0∼p0X_0 p_0 and choose (τ)=(1−τ)0+τ,∀τ∼Unif[0,1]X(τ)=(1-τ)X_0+ , ∀τ [0,1] Then, the corresponding conditional vector field is v(τ,(τ)|,χ)=−0.v(τ,X(τ)\,|\,Z,χ)=Z-X_0. At τ in the generative process, the estimated endpoint is ^=(τ)+(1−τ)fθ(τ,(τ)|χ). Z=X(τ)+(1-τ)f^θ(τ,X(τ)\,|\,χ). Let c^i,tb c_i,t^b and ξ^B,tb ξ_B,t^b denote the velocity of the contact point and the body twist of the plank obtained by decoding Z, and let ci,tb,∗c_i,t^b,* and ξB,tb,∗ _B,t^b,* denote the corresponding quantities from the trajectory used to construct Z. Let σc∈ℝ2 _c ^2 contain the training set scales for the two components of the contact point velocity, and let σξ∈ℝ3 _ξ ^3 contain the training set scales for the three components of the body twist of the plank. Define ρt=2.5 _t=2.5 for t=1,…,4t=1,…,4 and ρt=1 _t=1 otherwise. The loss function ℒMACL_ MAC for Task 1 is augmented as ℒtrain=ℒMAC+0.16ℒmotion+0.45ℒtwist,L_ train=L_ MAC+0.16\,L_ motion+0.45\,L_ twist, in which ℒmotion _ motion =12N∑t=1Tρt∑i=1N∑t=1Tρt = 12N _t=1^T _t _i=1^N _t=1^T _t ⋅‖diag(σc)−1(c^i,tb−ci,tb,∗)‖2 ·\, \|diag( _c)^-1 ( c_i,t^b-c_i,t^b,* ) \|^2 and ℒtwist=13∑t=1Tρt∑t=1Tρt‖diag(σξ)−1(ξ^B,tb−ξB,tb,∗)‖2.L_ twist= 13 _t=1^T _t _t=1^T _t \|diag( _ξ)^-1 ( ξ_B,t^b- _B,t^b,* ) \|^2. In addition, training and validation curves over three independent seeds are provided in Appendix B.3. B.2 Task and safety factors The task and safety factors are constructed from geometric margins. Each margin is positive when its corresponding requirement is satisfied. Specifically, the workspace is [Xmin,Xmax]×[Ymin,Ymax]=[−7,6.8]×[−3,3],[X_ ,X_ ]×[Y_ ,Y_ ]=[-7,6.8]×[-3,3], and the spatial gap takes x∈[xgapmin,xgapmax]=[0,2].x∈[x_ gap ,x_ gap ]=[0,2]. The plank length and width are LB=5.8L_B=5.8 and WB=0.72W_B=0.72, each robot is a disk with radius rR=0.16r_R=0.16, the required plank support on each side of the gap is dsup=0.55d_ sup=0.55, and the traversable region for the robot centers must extend at least dcor=0.05d_ cor=0.05 beyond each gap edge. Let gB,t=(RB,t,pB,t)g_B,t=(R_B,t,p_B,t), t=1,…,Texect=1,…,T_ exec, denote the plank poses predicted after the Texec=4T_ exec=4 actions that will be executed, with pB,t=[xt,yt]⊤p_B,t=[x_t,y_t] and orientation angle θt _t. Define |cos(θ)|ε≔cos2(θ)+ε2| (θ)|_ ^2(θ)+ ^2 and |sin(θ)|ε≔sin2(θ)+ε2| (θ)|_ ^2(θ)+ ^2 with ε=10−6 =10^-6. The distance from the plank center to its outermost point along the x-axis of the world frame is hx(θ)≔12(LB|cos(θ)|ε+WB|sin(θ)|ε).h_x(θ) 12\! (L_B| (θ)|_ +W_B| (θ)|_ ). The projection onto the same axis of the usable half-length of the plank for a robot center is hc(θ)≔(LB/2−rR−0.075)|cos(θ)|ε.h_c(θ) (L_B/2-r_R-0.075 )| (θ)|_ . At predicted step t, let the four margins be m1,ttask m_1,t task =xgapmin−xt+hx(θt)−dsup, =x_ gap -x_t+h_x( _t)-d_ sup, m2,ttask m_2,t task =xt+hx(θt)−xgapmax−dsup, =x_t+h_x( _t)-x_ gap -d_ sup, m3,ttask m_3,t task =xgapmin−xt+hc(θt)−dcor, =x_ gap -x_t+h_c( _t)-d_ cor, m4,ttask m_4,t task =xt+hc(θt)−xgapmax−dcor. =x_t+h_c( _t)-x_ gap -d_ cor. and let m5,ttask,…,mMtask,ttaskm_5,t task,…,m_M_ task,t task denote the remaining geometric margins included in the task factor, where MtaskM_ task is the total number of task margins. The task factor evaluates all of these margins at the final predicted plank pose as qtask(|χ)=180log(CLOSE q_ task(z\,|\,χ)= 180 ( ∑k=12exp[−80mk,Texectask0.55] _k=1^2 \! [-80\, m_k,T_ exec task0.55 ] +∑k=3Mtaskexp[−80mk,Texectask0.2]). + _k=3^M_ task \! [-80\, m_k,T_ exec task0.2 ] ). Let mt,1safe,…,mt,Msafesafem_t,1 safe,…,m_t,M_ safe safe denote the geometric margins at predicted step t∈1,…,Texect∈\1,…,T_ exec\, where MsafeM_ safe is the number of safety margins evaluated at each step. These margins measure the distances of the plank from the upper and lower workspace boundaries using a 0.20.2 buffer, the distances of the robot disks from the gap and from the upper and lower workspace boundaries using a 0.10.1 buffer, and whether the plank has sufficient support on the side of the workspace where robots are initiated or has reached sufficient support on the opposite side. The safety factor is then qsafe(|χ)=160log(∑t=1Texec∑k=1Msafeexp[−60mt,ksafe0.1]).q_ safe(z\,|\,χ)= 160 \! ( _t=1^T_ exec _k=1^M_ safe \! [-60\, m_t,k safe0.1 ] ). The success and failure are evaluated using the exact geometry (without buffers) instead of the factors above. A trial is successful if the robots can move the plank such that it provides at least dsup=0.55d_ sup=0.55 of support on both sides of the gap and the region traversable by the robot centers extends beyond both gap edges as required. A failure is counted if the plank loses support on the initial side before reaching the opposite side, leaves the workspace, or causes any robot to cross the workspace boundary or enter the spatial gap. For Task 1, we choose gi=I24g_i=I_24, Hi=I24H_i=I_24, and αaSE(s)=s _a SE(s)=s. For either factor, VaSE(τ,|χ)=qa(|χ)−βa(τ)V_a SE(τ,z\,|\,χ)=q_a(z\,|\,χ)- _a(τ). Hence, βa(τ) _a(τ) is the upper bound imposed on qaq_a during the generative process, and VaSE≤0V_a SE≤ 0 is equivalent to qa(|χ)≤βa(τ)q_a(z\,|\,χ)≤ _a(τ). Choose βa(τ)=βa,end+(βa,start−βa,end)(1−6τ5+15τ4−10τ3). _a(τ)= _a, end+ ( _a, start- _a, end )(1-6τ^5+15τ^4-10τ^3). At the beginning of each policy generation, let gBcurg_B cur denote the plank pose and let qtaskcurq_ task cur denote the task factor evaluated at gBcurg_B cur instead of the predicted terminal plank pose. Let 0≔(0)z_0 (0) denote the Gaussian initial state during the generative process. Specifically, βtask,end=qtaskcur−0.05, _ task,end=q_ task cur-0.05, βtask,start=maxqtask(0|χ)+0.02,βtask,end+0.02, _ task,start= \! \q_ task(z_0\,|\,χ)+0.02,\, _ task,end+0.02 \, βsafe,end=0, _ safe,end=0, βsafe,start=qsafe(0|χ)+0.02. _ safe,start=q_ safe(z_0\,|\,χ)+0.02. Therefore, at the end of each generative process, the task factor is required to satisfy qtask≤qtaskcur−0.05q_ task≤ q_ task cur-0.05, and the safety factor is required to satisfy qsafe≤0q_ safe≤ 0. Note that βtask,start _task, start is calculated once before the generative process and is fixed throughout the generative process so βtask(τ) _ task(τ) is C1C^1 in τ. In addition, the adaptive constraint allocation coefficients wa,iSEw_a,i SE are calculated using (14) with δ=0.02δ=0.02. Each robot then solves the decoupled guidance using (15) with one constraint corresponding to the task factor and one corresponding to the safety factor. B.3 Additional results for Task 1 Figure 5 shows an N=7N=7 trial in which the two policies, Nominal and Guided, start from the same physical configuration and initial sample. The policy generated by the nominal vector field drives the plank out of the workspace, whereas the policy generated with SE guidance avoids this failure and successfully constructs the bridge. Figure 6 shows the evolution of qsafeq_ safe and its time-varying upper bound βsafe _ safe during a generative process with SE guidance right before the policies generated by the nominal vector field drive the plank out of the workspace. During the generative process, qsafe((τ)|χ)q_ safe(z(τ)\,|\,χ) stays below βsafe(τ) _ safe(τ) as the latter goes to βsafe(1)=0 _ safe(1)=0, with the generated policy satisfying qsafe((1)|χ)<0q_ safe(z(1)\,|\,χ)<0. Figure 5: An N=7N=7 trial of Task 1. Left: the policies generated by the nominal vector field are driving the plank out of the workspace. Right: the policies with SE guidance lead to successful bridge construction. Figure 6: Evolution of qsafeq_ safe and its time-varying upper bound βsafe _ safe during a generative process with SE guidance at a physical time right before the policies generated by the nominal vector field drive the plank out of the workspace in an N=7N=7 trial. Figure 7 shows the training and validation losses over three independent training seeds. The lines show the mean across seeds, and the shaded regions show ± one sample standard deviation. Figure 7: Training and validation losses over three independent seeds for the GNN-represented nominal vector field in Task 1. Shaded regions represent ± one sample standard deviation. Appendix C Task 2: Detailed Settings and Additional Results C.1 Detailed settings for Task 2 Task 2 considers tabletop scenes with a library of 12 object types. For each scene, N different objects are randomly selected from the library. Each object has a fixed geometric shape and one or more nearby regions that should remain unobstructed when the object is used, which are referred to as affordance regions. A user-specified affordance requirement can change the location, size, or direction of these regions relative to the object. An object can therefore be required to remain accessible from a different side, to have more free space around it, or to satisfy combinations of these changes. The complete object library, geometric representations, affordance-region definitions, and ranges used to generate modified requirements are given below. For Task 2, choose gi=I3g_i=I_3, δPE=0.02δ^PE=0.02, ωPE(τ)=0.1+1.8τ,ω^PE(τ)=0.1+1.8τ, cPE=1.15(1−ρPE)maxVPE(0),10−301−ρPE,c^PE= 1.15(1-ρ^PE) \! \V^PE(0),10^-30 \^1-ρ^PE, and ρPE=0.75ρ^PE=0.75. The tabletop is centered at the origin and has size 1.40×0.941.40× 0.94. For each object i, zi∈ℝ3z_i ^3 is decoded into a planar position pi∈ℝ2p_i ^2 and an angular coordinate θi∈ℝ _i by [pi⊤,θi]⊤=diag(0.70,0.47,1.35)zi[p_i , _i] =diag(0.70,0.47,1.35)\,z_i with the corresponding orientation Ri(θi)∈(2)R_i( _i) (2). Each type of object also has a fixed geometric shape defined in its body frame, represented as a disk, a rounded rectangle, or a combination of these shapes. Table 3 summarizes the 12 object categories, with each category showing at most once in a scene. Denote C(r)C(r) for a disk of radius r, R(a,b,rc)R(a,b,r_c) for a rounded rectangle with half width a, half height b, and rounded corner radius rcr_c, and A(x,y,r)A(x,y,r) for a circular affordance region of radius r centered at (x,y)(x,y) in the object’s body frame. For multiple affordance regions with the same y-coordinate and radius, we list their x-coordinates as a set. For example, A(x1,…,xm,y,r)A(\x_1,…,x_m\,y,r) denotes circles centered at (x1,y),…,(xm,y)(x_1,y),…,(x_m,y), each with radius r. For the qqth affordance region of object i, let aiq∈ℝ2a_iq ^2 denote the center of the region in the body frame of object i, and let riq>0r_iq>0 denote its radius. A user-specified requirement is parameterized by (sio,sir,φi)(s_i^o,s_i^r, _i), where sios_i^o changes the distance of the affordance region from the object, sirs_i^r changes its radius, and φi _i changes its direction relative to the object. Then, the center and radius expressed in the world frame are aiqW=pi+R(θi+φi)sioaiqa_iq^W=p_i+R( _i+ _i)s_i^oa_iq and riqW=sirriqr_iq^W=s_i^rr_iq, respectively. The default requirement corresponds to (sio,sir,φi)=(1,1,0)(s_i^o,s_i^r, _i)=(1,1,0). A user may instead require an object to be accessible from another side, require more free space around the object, or combine these changes. We sample sio∈[0.98,1.12]s_i^o∈[0.98,1.12] and sir∈[1.04,1.24]s_i^r∈[1.04,1.24]. For a change in access direction, φi _i is selected from the allowed directions for that object type, with an additional perturbation in [−π/22.5,π/22.5][-π/22.5,π/22.5]. The alternative directions are −π/6-π/6 and π/6π/6 for drawer organizer and drawing board; −π/2-π/2 and π/2π/2 for desk lamp, handled mug, and tissue box; π, −π/2-π/2, and π/2π/2 for teacup, laptop, scissors, pencil case, and pen cup; and −π/2-π/2, π/2π/2, and π for potted plant and compact printer. Table 3: Object geometry and default affordance regions for Task 2 Object Geometry Affordance region(s) Drawer organizer R(0.156,0.090,0.012)R(0.156,0.090,0.012) A(−0.1,0,0.1,−0.205,0.108)A(\-0.1,0,0.1\,-0.205,0.108) Teacup C(0.046)∪(C(0.019)+(0.061,0))C(0.046)∪(C(0.019)+(0.061,0)) A(0.125,0,0.055)A(0.125,0,0.055) Desk lamp C(0.06)C(0.06) A(0,−0.147,0.079)A(0,-0.147,0.079) Laptop R(0.085,0.065,0.006)R(0.085,0.065,0.006) A(0.155,0,0.07)A(0.155,0,0.07) Potted plant C(0.078)C(0.078) A(0,−0.123,0.062)A(0,-0.123,0.062) Handled mug C(0.067)C(0.067) A(0,−0.141,0.054)A(0,-0.141,0.054) Drawing board R(0.135,0.048,0.008)R(0.135,0.048,0.008) A(−0.065,0.065,−0.108,0.05)A(\-0.065,0.065\,-0.108,0.05) Scissors R(0.039,0.026,0.016)R(0.039,0.026,0.016) A(0.083,0,0.043)A(0.083,0,0.043) Compact printer R(0.099,0.07,0.009)R(0.099,0.07,0.009) A(0,−0.133,0.057)A(0,-0.133,0.057) Pencil case R(0.037,0.072,0.007)R(0.037,0.072,0.007) A(0.087,0,0.046)A(0.087,0,0.046) Tissue box R(0.049,0.04,0.009)R(0.049,0.04,0.009) A(0,−0.103,0.047)A(0,-0.103,0.047) Pen cup C(0.034)C(0.034) A(0.083,0,0.039)A(0.083,0,0.039) In addition, training contains 4,000 scenes for each N∈4,5,6N∈\4,5,6\ and 400 validation scenes for each N. For every scene, N different types of objects are sampled uniformly without replacement from the library. Thus, all object categories appear during training, while N=7,8N=7,8 evaluate generalization to scenes containing more objects than those used for training. The nominal vector field is represented by a GNN shared by every agent. It is trained using (5) without augmentation. C.2 Additional results for Task 2 We additionally evaluate five types of requirement: the default requirements, opposite-side access, side-direction access, a larger affordance region, and combinations of these changes. Opposite-side access selects the direction farthest from the default one, and side-direction access selects one of the other non-default directions. Each type contains 50 cases, with 10 cases for each N∈4,5,6,7,8N∈\4,5,6,7,8\. As shown in Table 4, Guided directly generates the scenes satisfying all 50 cases in every type of requirement without retraining the nominal vector field. Table 4: Task 2 results with different affordance requirements. Requirement Natural Nominal Private Nominal Guided Default 27/5027/50 (54%) 27/5027/50 (54%) 50/5050/50 (100%) Opposite-side access 27/5027/50 (54%) 23/5023/50 (46%) 50/5050/50 (100%) Side-direction access 31/5031/50 (62%) 23/5023/50 (46%) 50/5050/50 (100%) Larger affordance region 23/5023/50 (46%) 22/5022/50 (44%) 50/5050/50 (100%) Combined 27/5027/50 (54%) 17/5017/50 (34%) 50/5050/50 (100%) Figure 8: An N=8N=8 example of Task 2. Green circles denote unobstructed affordance regions, a red circle denotes an obstructed affordance region, and red object outlines indicate pairs of objects that are too close to each other. Left: the compact printer and scissors, and the desk lamp and pen cup, are too close to each other, and the drawer organizer obstructs the affordance region of the handled mug. Right: PE guidance resolves all three violations from the same initial sample without any model retraining. Figure 8 shows a complementary N=8N=8 example without any user-specified change to the default requirements. The scene generated by the nominal vector field contains several violations: the compact printer and scissors are too close to each other, the desk lamp and pen cup are also too close to each other, and the drawer organizer obstructs the affordance region of the handled mug. Starting from the same initial Gaussian sample, PE guidance directly generates a scene satisfying all of these requirements, without any model retraining. Figure 9 shows the training and validation flow matching losses over three independent training seeds. The lines show the mean across seeds, and the shaded regions show ± one sample standard deviation. We further evaluate how much PE guidance changes the distribution characterized by the nominal flow. Task 2 provides a natural setting for this evaluation as each generative process directly produces one complete scene consisting of the poses of all N objects. For each fixed N∈4,5,6,7,8N∈\4,5,6,7,8\, let M=50M=50 denote the number of evaluated scenes. For scene m∈1,…,Mm∈\1,…,M\ and object i∈1,…,Ni∈\1,…,N\, denote the final generated states by Nominal and Guided by zm,iNomz_m,i Nom and zm,iGuiz_m,i Gui, respectively. The corresponding joint generative states of the complete scenes are then denoted by mNom=[(zm,1Nom)⊤,…,(zm,NNom)⊤]⊤z_m Nom=[(z_m,1 Nom) ,…,(z_m,N Nom) ] and mGui=[(zm,1Gui)⊤,…,(zm,NGui)⊤]⊤z_m Gui=[(z_m,1 Gui) ,…,(z_m,N Gui) ] , respectively. Thus, the two sets mNomm=1M\z_m Nom\_m=1^M and mGuim=1M\z_m Gui\_m=1^M approximate the corresponding Nominal and Guided scene distributions. First, we evaluate the distance directly in the generative state space. For two scene states =[z1⊤,…,zN⊤]⊤z=[z_1 ,…,z_N ] and ′=[(z1′)⊤,…,(zN′)⊤]⊤z =[(z_1 ) ,…,(z_N ) ] , denote the normalized Euclidean distance as dz2(,′)=1N∑i=1N‖zi−zi′‖22d_z^2(z,z )= 1N _i=1^N\|z_i-z_i \|_2^2. Using the M generated scenes, we approximate the Wasserstein distance between the Nominal and Guided joint distributions as W^2,z 2(N)=minσ∈M1M∑m=1Mdz2(mNom,σ(m)Gui), W_2,z^\,2(N)= _σ∈ S_M 1M _m=1^Md_z^2 (z_m Nom,z_σ(m) Gui ), where M S_M denotes the set of all permutations of 1,…,M\1,…,M\. Thus, σ specifies a optimal transport assignment between the M Nominal scenes and the M Guided scenes. We also evaluate the same distributional deviation after decoding into object poses using [(pm,ib)⊤,θm,ib]⊤=diag(0.70,0.47,1.35)zm,ib[(p_m,i^b) , _m,i^b] =diag(0.70,0.47,1.35)\,z_m,i^b, with b∈Nom,Guib∈\ Nom, Gui\ representing Nominal and Guided, and gm,ib∈(2)g_m,i^b (2). Denote the complete decoded scene as mb=(gm,1b,…,gm,Nb)g_m^b= (g_m,1^b,…,g_m,N^b ). For two object poses gig_i and gi′g_i , we use the geodesic (Park, 1995) d(2)2(gi,gi′)=‖pi−pi′‖22+λθ2Δθi2,d_SE(2)^2(g_i,g_i )=\|p_i-p_i \|_2^2+ _θ^2 _i^2, where Δθi=atan2(sin(θi−θi′),cos(θi−θi′))∈[−π,π] _i=atan2 ( ( _i- _i ), ( _i- _i ) )∈[-π,π], and we choose λθ=0.1 _θ=0.1, which places translational and rotational differences on a common length scale comparable to the sizes of the objects in Table 3. For two decoded scenes =(g1,…,gN)g=(g_1,…,g_N) and ′=(g1′,…,gN′)g =(g_1 ,…,g_N ), define the scene distance by dscene2(,′)=1N∑i=1Nd(2)2(gi,gi′).d_ scene^2(g,g )= 1N _i=1^Nd_SE(2)^2(g_i,g_i ). Then the corresponding approximation of the Wasserstein distance is W^2,g 2(N)=minσ∈M1M∑m=1Mdscene2(mNom,σ(m)Gui). W_2,g^\,2(N)= _σ∈ S_M 1M _m=1^Md_ scene^2 (g_m Nom,g_σ(m) Gui ). The magnitude of a Wasserstein distance is difficult to interpret without a reference scale, so we compare it with the root mean square (RMS) pairwise distance among the M complete scenes generated by the nominal vector field. In the generative state space, define SNom,z(N)=(2M(M−1)∑1≤m<n≤Mdz2(mNom,nNom))12.S_ Nom,z(N)= (\! 2M(M-1) _1≤ m<n≤ M\!\!d_z^2 (z_m Nom,z_n Nom )\! )^\!\! 12\!. Similarly, in the decoded physical space, define SNom,g(N)=(2M(M−1)∑1≤m<n≤Mdscene2(mNom,nNom))12.S_ Nom,g(N)= (\!\! 2M(M-1)\! _1≤ m<n≤ M\!\!\!d_ scene^2 (g_m Nom,g_n Nom )\!\! )^\!\!\! 12\!\!. We then plot the two ratios W^2,z(N)SNom,z(N) W_2,z(N)S_ Nom,z(N) and W^2,g(N)SNom,g(N) W_2,g(N)S_ Nom,g(N) in Figure 10, where the first ratio measures the Wasserstein distance between the Nominal and Guided distributions in the generative state space relative to the RMS pairwise distance among the M complete scenes generated by the nominal vector field, and the second ratio measures the one in the decoded pose space. For example, a value of 0.30.3 means that the Wasserstein distance between the Nominal and Guided distributions is 0.30.3 times the RMS pairwise distance among the M complete scenes generated by the nominal vector field. As the number of objects increases, both curves increase gradually, meaning that guidance makes larger modifications when there are more objects and thus more coupled requirements. As shown in Figure 10, in the generative state space, W^2,z(N)/SNom,z(N) W_2,z(N)/S_ Nom,z(N) increases from 20.6%20.6\% for N=4N=4 to 34.5%34.5\% for N=8N=8. In the decoded pose space, W^2,g(N)/SNom,g(N) W_2,g(N)/S_ Nom,g(N) increases from 20.5%20.5\% to 33.7%33.7\%. Thus, even for N=8N=8, the Wasserstein distance between the Nominal and Guided distributions is only about one third of the RMS pairwise distance among the complete scenes generated by the nominal vector field that measures how spread out the generated scenes are. The results show that guidance introduce a distributional deviation that is much smaller than the variation among scenes generated by the nominal vector field. We additionally examine whether guidance reduces how spread the generated objects are. Define SGui,z(N)=(2M(M−1)∑1≤m<n≤Mdz2(mGui,nGui))12,S_ Gui,z(N)= ( 2M(M-1) _1≤ m<n≤ Md_z^2 (z_m Gui,z_n Gui ) ) 12, and SGui,g(N)=(2M(M−1)∑1≤m<n≤Mdscene2(mGui,nGui))12.S_ Gui,g(N)= (\! 2M(M-1) _1≤ m<n≤ M\!\!d_ scene^2 (g_m Gui,g_n Gui )\! )^\!\! 12\!\!. These quantities are defined in exactly the same way as SNom,z(N)S_ Nom,z(N) and SNom,g(N)S_ Nom,g(N), but using the M complete scenes generated with guidance. Across N=4,…,8N=4,…,8, SGui,z(N)/SNom,z(N)S_ Gui,z(N)/S_ Nom,z(N) ranges from 98.3%98.3\% to 99.9%99.9\%, and SGui,g(N)/SNom,g(N)S_ Gui,g(N)/S_ Nom,g(N) ranges from 99.1%99.1\% to 100.5%100.5\%, meaning that the RMS pairwise distance among Guided scenes is almost the same as that among scenes generated by the nominal vector field. These results show that the generated objects with PE guidance satisfy the desired hard requirements while leading to only a moderate deviation of the distribution resulting from the nominal vector field, and approximately preserving the spread of the generated scenes. Figure 9: Training and validation flow matching losses over three independent seeds for the nominal vector field in Task 2. Shaded regions represent ± one sample standard deviation. Figure 10: Approximation of the Wasserstein distance between the Nominal and Guided distributions in Task 2 with respect the total number of generated objects, normalized by the RMS pairwise distance among the M=50M=50 scenes generated by the nominal vector field. The blue and orange curves use the distance metrics in the generative state space and the decoded pose space, respectively.