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Multi-Agent System Identification with Nonlinear Sheaf Diffusion
Nivar Anwer, Hans Riess, Matthew Hale
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 89%
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Summary
This paper investigates the identifiability of local interaction laws in multi-agent systems governed by nonlinear sheaf diffusion. It establishes that the fundamental obstruction to recovering edge potentials from trajectory data is topological, characterized by the first sheaf cohomology group. Unique recovery is possible if and only if this cohomology vanishes. For nontrivial cohomology, identifiability is restored within finite-dimensional parameterized classes when a data-dependent information matrix is positive definite. Experiments validate these theoretical findings across various coordination dynamics.
Entities (10)
Relation Signals (8)
Multi-agent systems → governedby → Nonlinear Sheaf Laplacian
confidence 95% · In systems governed by a nonlinear sheaf Laplacian -- a generalization of the graph Laplacian accommodating heterogeneous state spaces and asymmetric communication channels
Sheaf Cohomology H1 → measures → Obstruction to Recovery
confidence 95% · the fundamental obstruction to recovery is topological, measured by sheaf cohomology, and that unique recovery from an unconstrained function class is possible if and only if this cohomology vanishes.
Information Matrix → enables → Recovery in Parameterized Class
confidence 90% · recovery within a finite-dimensional parameterized class is possible precisely when a data-dependent information matrix is positive definite.
Trajectory Data → exposes → Aggregate Effect of Edge Forces
confidence 90% · Because trajectory observations record node-state evolution, they expose only the aggregate effect of the edge forces at each node
Nonlinear Sheaf Laplacian → generalizes → Graph Laplacian
confidence 90% · a generalization of the graph Laplacian accommodating heterogeneous state spaces and asymmetric communication channels
Edge Potential Functions → generates → Inter-agent Forces
confidence 90% · the coordination law is encoded by edge potential functions whose gradients produce the inter-agent forces.
Cellular Sheaves → providesframeworkfor → Nonlinear Sheaf Laplacian
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Abstract
Abstract:Local interaction laws governing multi-agent systems can be difficult to recover from trajectory data, even when the dynamics are observed faithfully. In systems governed by a nonlinear sheaf Laplacian -- a generalization of the graph Laplacian accommodating heterogeneous state spaces and asymmetric communication channels -- the coordination law is encoded by edge potential functions whose gradients produce the inter-agent forces. Because trajectory observations record node-state evolution, they expose only the aggregate effect of the edge forces at each node: distinct interaction laws that agree at the node level are indistinguishable from trajectory data alone. We show that the fundamental obstruction to recovery is topological, measured by sheaf cohomology, and that unique recovery from an unconstrained function class is possible if and only if this cohomology vanishes. When the obstruction is nontrivial, we show that recovery within a finite-dimensional parameterized class is possible precisely when a data-dependent information matrix is positive definite. Experiments validate the theory and illustrate that accurate trajectory reproduction need not certify recovery of the underlying interaction law.
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- Source: https://arxiv.org/abs/2605.11204v1
- Canonical: https://arxiv.org/abs/2605.11204v1
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Multi-Agent System Identification with Nonlinear Sheaf Diffusion Nivar Anwer, Hans Riess, and Matthew Hale Anwer is with the Department of Computer Science, Georgia Tech. Riess and Hale are with the Department of Electrical & Computer Engineering, Georgia Tech. Riess was supported by DARPA (HR0011-25-3-0235); Riess and Hale were supported by AFOSR (FA9550-23-1-0120, FA9550-19-1-0169) and ONR (N00014-22-1-2435). Emails: nanwer3,riess,mhale30@gatech.edu. Abstract Local interaction laws governing multi-agent systems can be difficult to recover from trajectory data, even when the dynamics are observed faithfully. In systems governed by a nonlinear sheaf Laplacian—a generalization of the graph Laplacian accommodating heterogeneous state spaces and asymmetric communication channels—the coordination law is encoded by edge potential functions whose gradients produce the inter-agent forces. Because trajectory observations record node-state evolution, they expose only the aggregate effect of the edge forces at each node: distinct interaction laws that agree at the node level are indistinguishable from trajectory data alone. We show that the fundamental obstruction to recovery is topological, measured by sheaf cohomology, and that unique recovery from an unconstrained function class is possible if and only if this cohomology vanishes. When the obstruction is nontrivial, we show that recovery within a finite-dimensional parameterized class is possible precisely when a data-dependent information matrix is positive definite. Experiments validate the theory and illustrate that accurate trajectory reproduction need not certify recovery of the underlying interaction law. I Introduction In multi-agent systems (MAS) encompassing robotic swarms, opinion networks, and distributed estimation, collective behavior emerges from local interaction laws—rules specifying what neighboring agents compare and how disagreement is converted into a coordinating force. The classical framework, grounded in spectral graph theory [chungSpectralGraphTheory1997, saberAgreementProblemsNetworks2003, OlfatiSaber2007, Oh2015], models these laws via a graph Laplacian acting uniformly on a shared state space. Cellular sheaves [currySheavesCosheavesApplications2014] and their spectral theory [hansenSheafLaplacianSpectral2019] extend this classical framework to the heterogeneous setting: a sheaf over a graph assigns distinct local state spaces and shared comparison spaces to vertices and edges (respectively), with restriction maps specifying how agent states project into communication channels. The nonlinear sheaf Laplacian, formed by composing these restriction maps with edge potentials, subsumes consensus, formation control, opinion dynamics, and target tracking within a single operator [hanksDistributedMultiAgentCoordination2025, zhaoAsynchronousNonlinearSheaf2025, hanksHeterogeneousMultiAgentMultiTarget2025]. Sheaf Laplacians have also attracted growing attention in graph learning and structural causal models [bodnarNeuralSheafDiffusion2022, battiloroTangentBundleConvolutional2024, zaghenSheafDiffusionGoes2024, hansenGebhartSheafNeuralNetworks2020, dacuntoLearningConsistentCausal2026]—works that learn the sheaf operator for a downstream task, in contrast to our inverse problem of recovering a fixed edge potential from trajectory data. In this paper, we study the inverse problem: when can the local interaction law be recovered by observing trajectories of a MAS? A growing literature addresses this question for interacting particle systems [Lu2019, Li2021Identifiability, luMaggioniTangHeterogeneous2021, Miller2023], establishing identifiability and consistency under coercivity or excitation conditions, yet predominantly assuming homogeneous interactions over a shared state space. A related line of work learns topological structure from data: Hansen & Ghrist learn sheaf Laplacians from smooth signals [hansenLearningSheafLaplacians2019, dininoLearningStructureConnection2026], while other approaches recover graph topology from observed dynamics [segarraNetworkInferenceConsensus2017, zhuConsensusUnknown2020, talukdarPhysicsInformedTopology2020]. While the first sheaf cohomology is well understood as an obstruction to coordination [hansenOpinionDynamicsDiscourse2021], its role as an obstruction to recovering a nonlinear edge potential from trajectory data in a dynamical MAS setting has not yet been characterized. The present paper addresses this gap. Our contributions are as follows. 1. We show that the fundamental obstruction to edge-law recovery from trajectory data is topological: the space of edge forces invisible to node-level observations coincides with kerδ∗ δ^*, which by the Hodge decomposition is isomorphic to the first sheaf cohomology H1(G;ℱ)H^1(G;F). Recovery from an unconstrained function class is possible if and only if H1(G;ℱ)=0H^1(G;F)=0. 2. When H1(G;ℱ)≠0H^1(G;F)≠ 0, we prove that identifiability is restored within a finite-dimensional parameterized class whenever a data-dependent information matrix is positive definite. 3. Experiments on formation transfer, bounded-confidence dynamics, and finite-basis models validate the theory across all three failure modes: cohomological ambiguity, weak excitation, and basis rank deficiency. The remainder of the paper is organized as follows. Section I introduces Euclidean sheaves, the nonlinear sheaf Laplacian, and the system identification problem. Section I establishes the fundamental limitations of nonparametric recovery. Section IV develops the parametric identifiability criterion and least-squares estimator. Section V presents experiments, and Section VI concludes. I Preliminaries & Problem Formulation We introduce the background material on cellular sheaves of inner-product spaces over graphs and the nonlinear sheaf Laplacian. I-A Euclidean Sheaves Suppose G=(V,E)G=(V,E) is a directed graph with finite vertex-set V, finite edge-set E, and maps h,t:E⇉Vh,t:E V sending an edge e to its head h(e)h(e) or tail t(e)t(e). We define a partially ordered set (G)≔(V⊔E,⊴) P(G) (V E, ) on the disjoint union generated by the relation: v⊴ev e whenever e∈h−1(v)∪t−1(v)e∈ h^-1(v)∪ t^-1(v). This partial order encodes incidence relations between vertices and edges of the graph and will be utilized to index spaces and linear maps in the following definition. Definition 1. A Euclidean sheaf over G=(V,E)G=(V,E) is an assignment of the following to G: Stalks : A real inner-product space ℱ(v)F(v) to every v∈Vv∈ V with ⟨xv,xv′,⟩v≔xv⊤Rvxv′ x_v,x_v , _v x_v R_vx_v , Rv≻0R_v 0. Edge stalks : A real inner-product space ℱ(e)F(e) to every e∈Ee∈ E with ⟨ye,ye′,⟩e≔ye⊤Reye′ y_e,y_e , _e y_e R_ey_e , Re≻0R_e 0. Restriction maps : A pair of bounded linear maps ℱ(h(e))→ℱh(e)⊴eℱ(e),ℱ(t(e))→ℱt(e)⊴eℱ(e) (h(e)) F_h(e) eF(e), (t(e)) F_t(e) eF(e) (2) for every e∈Ee∈ E. Interpretation 1. In a MAS, G=(V,E)G=(V,E) models the communication or flow of information between agents: V indexes the agents of the system and E indexes communication channels between pairs of agents. ℱ(v)F(v) is the local state space of an agent v∈Vv∈ V. ℱ(e)F(e) is a shared communication space for the edge e∈Ee∈ E. The restriction map ℱv⊴eF_v e models the transmission of information about the state of an agent to a channel. The positive-definite matrices RvR_v and ReR_e define weighted inner products on the stalk and edge-stalk spaces, allowing different coordinates to be weighted according to their relative importance in the communication channel. Definition 2. Suppose ℱF is a Euclidean sheaf over G. Let C0(G;ℱ)≔⨁v∈Vℱ(v),C1(G;ℱ)≔⨁e∈Eℱ(e) C^0(G;F) _v∈ VF(v), C^1(G;F) _e∈ EF(e) (3) denote the spaces of 0-cochains and 1-cochains, respectively, with inner products ⟨x,x′⟩C0≔∑v∈V⟨xv,xv′⟩v,⟨y,y′⟩C1≔∑e∈E⟨ye,ye′⟩e x,x _C^0 _v∈ V x_v,x _v _v, y,y _C^1 _e∈ E y_e,y _e _e (4) The coboundary map of ℱF is a bounded linear map δℱ:C0(G;ℱ)→C1(G;ℱ) _F:C^0(G;F)→ C^1(G;F) defined by (δℱx)e≔ℱh(e)⊴exh(e)−ℱt(e)⊴ext(e).( _Fx)_e F_h(e) ex_h(e)-F_t(e) ex_t(e). (5) The kernel kerδℱ⊆C0(G;ℱ) _F C^0(G;F) is the space of global sections H0(G;ℱ)H^0(G;F). The first sheaf cohomology is H1(G;ℱ):=C1(G;ℱ)/imδℱH^1(G;F):=C^1(G;F)/im\, _F. Notation 1. In coordinates, δℱ _F (written δ when ℱF is clear from context) is represented by a d1×d0d_1× d_0 matrix B, and its Hilbert-space adjoint satisfies δ∗=M1−1B⊤M2δ =M_1^-1B M_2, where M1≻0M_1 0 and M2≻0M_2 0 are the block-diagonal Gram matrices of C0(G;ℱ)C^0(G;F) and C1(G;ℱ)C^1(G;F), respectively. Remark 1. H1(G;ℱ)H^1(G;F) is a quotient vector space whose elements are equivalence classes [y]=y+imδℱ[y]=y+im\, _F. Interpretation 2. In a MAS, C0(G;ℱ)C^0(G;F) is the (global) state space of the system. δℱ _F simultaneously measures the disagreements between all agents in communication channels. H0(G;ℱ)H^0(G;F) collects global states x∈C0(G;ℱ)x∈ C^0(G;F) that are in agreement, i.e. ℱh(e)⊴e(xh(e))=ℱt(e)⊴e(xt(e))F_h(e) e(x_h(e))=F_t(e) e(x_t(e)). H1(G;ℱ)H^1(G;F) measures obstructions to agreement. A Hodge Decomposition [eckmannHarmonischeFunktionenUnd1944] applies to Euclidean sheaves over graphs, implying the following lemma. Lemma 1 (Hodge decomposition, [hansenSheafLaplacianSpectral2019]). C1(G;ℱ)=imδ⊕kerδ∗C^1(G;F)=im\,δ \,δ^*, so kerδ∗≅H1(G;ℱ) \,δ^* H^1(G;F) as inner-product spaces. I-B Dynamical Systems on Sheaves We now turn to dynamical systems governed by a Laplace operator defined for a given sheaf over G. Definition 3. Suppose Φ:C1(G;ℱ)→C1(G;ℱ) :C^1(G;F)→ C^1(G;F) is bounded and continuous. The nonlinear sheaf Laplacian is the operator LℱΦ:C0(G;ℱ)→C0(G;ℱ)L _F:C^0(G;F)→ C^0(G;F) defined by LℱΦ≔δ∗∘Φ∘δL _F δ δ. The linear sheaf Laplacian Lℱ:=δ∗∘δL_F:=δ^* δ (corresponding to Φ=id =id) was introduced and studied in [hansenSheafLaplacianSpectral2019]. Solutions of x˙=−Lℱx x=-L_Fx converge exponentially to the orthogonal projection of x(0)x(0) onto kerLℱ=H0(G;ℱ) L_F=H^0(G;F) [hansenSheafLaplacianSpectral2019, Proposition 8.1]. The nonlinear case was first studied in [hansenOpinionDynamicsDiscourse2021], which establishes convergence to H0(G;ℱ)H^0(G;F) for the special case of odd, monotone Φ [hansenOpinionDynamicsDiscourse2021, Proposition 10.1]. We consider a more general assumption about Φ . Assumption 1. There exists U:C1(G;ℱ)→ℝU:C^1(G;F) such that ∇U=Φ∇ U= . Moreover, U:C1(G;ℱ)→ℝU:C^1(G;F) can be written as a sum U(y)=∑e∈EUe(ye)U(y)= _e∈ EU_e(y_e). Under 1, we have the following convergence result for the nonlinear sheaf diffusion equation. Theorem 1 ([hanksDistributedMultiAgentCoordination2025, Theorem 2]). Let ℱF be a Euclidean sheaf on G and suppose each edge potential Ue:ℱ(e)→ℝU_e:F(e) is strongly convex with unique minimizer be∈ℱ(e)b_e (e). Then for any diffusivity α>0α>0, trajectories of x˙=−αLℱ∇U(x) x=-α\,L_F^∇ U(x) (6) converge to the orthogonal projection of x(0)x(0) onto δℱ+b+H0(G;ℱ)=kerLℱ∇U, _F^+b+H^0(G;F)= L_F^∇ U, (7) where δℱ+:=(δℱ∗δℱ)−1δℱ∗ _F^+:=( _F^* _F)^-1 _F^* denotes the Moore–Penrose pseudoinverse of δℱ _F (restricted to (kerδℱ)⟂( _F) ). I-C Problem Formulation We now formulate our system identification problem for recovering the edge potentials from trajectories of sheaf dynamics. The sheaf diffusion equation is the ODE x˙=−LℱΦ(x)−Ψ(x) x=-L_F (x)- (x) (8) where Φ (edge field) and Ψ (node field) satisfy 1 and 2. Assumption 2. There exists W:C0(G;ℱ)→ℝW:C^0(G;F) such that ∇W=Ψ∇ W= , and W decomposes as W(x)=∑v∈VWv(xv)W(x)= _v∈ VW_v(x_v). The following example collects noteworthy edge potential functions, including the laws studied in our experiments in Section IV. Example 1 (Edge potentials). The following classes arise in coordination and opinion dynamics [hansenOpinionDynamicsDiscourse2021, hanksDistributedMultiAgentCoordination2025]: Quadratic : Ue(ye)=12‖ye‖ℱ(e)2U_e(y_e)= 12\|y_e\|^2_F(e), giving Φ=id =id and recovering the linear sheaf Laplacian LℱL_F. Shifted quadratic (formation control) : Ue(ye)=12‖ye−y^e‖ℱ(e)2U_e(y_e)= 12\|y_e- y_e\|^2_F(e) for a target 1-cochain y^∈C1(G;ℱ) y∈ C^1(G;F). The force ∇Ue(ye)=ye−y^e∇ U_e(y_e)=y_e- y_e drives the system toward δx=y^δ x= y; when y^∈imδ y \,δ this corresponds to a prescribed formation offset p^ij p_ij between neighbors i and j [hanksDistributedMultiAgentCoordination2025]. Bounded confidence (opinion dynamics) : Ue(ye)=ψe(‖ye‖ℱ(e)2)U_e(y_e)= _e(\|y_e\|^2_F(e)) for a smooth ψe:[0,∞)→ℝ _e:[0,∞) with ψe′(z)>0 _e (z)>0 for z<Dez<D_e and ψe′(z)=0 _e (z)=0 for z≥Dez≥ D_e. Agents communicate only when disagreement is below threshold DeD_e, reproducing Hegselmann–Krause bounded-confidence dynamics on the sheaf [hansenOpinionDynamicsDiscourse2021]. Antagonistic : Ue(ye)=−‖ye‖ℱ(e)2U_e(y_e)=-\|y_e\|^2_F(e) for edges in a set E−⊆E_- E of adversarial links. The resulting signed Laplacian LℱS=δ∗SδL^S_F=δ^*Sδ (with S=−IS=-I on E−E_-, S=IS=I on E+E_+) may be indefinite, producing unstable dynamics when E−E_- is a cutset [hansenOpinionDynamicsDiscourse2021]. With this family of examples in view, we now formulate the central inverse problem: recovering the edge potential from trajectory observations. Problem 1 (System Identification). Given trajectory data x(i)(t)i=1M\x^(i)(t)\_i=1^M from solutions of (8) under Assumptions 1–2, recover the edge potential U:C1(G;ℱ)→ℝU:C^1(G;F) , or characterize the obstructions to doing so. I Non-Parametric Potential Reconstruction The SI problem reduces to recovering the edge force law Φ=∇U:C1(G;ℱ)→C1(G;ℱ) =∇ U:C^1(G;F)→ C^1(G;F) from trajectory observations. Throughout this section, we assume Ψ satisfies Assumption 2 with each node potential WvW_v given as prior knowledge, so that the residual r(x):=−x˙−Ψ(x)=δ∗Φ(δx)r(x):=- x- (x)=δ^* (δ x) (9) is directly computable from trajectory data. Suppose trajectories x(i):[0,Ti]→C0(G;ℱ)x^(i):[0,T_i]→ C^0(G;F), i=1,…,Mi=1,…,M, are sampled from the dynamical system (8) with unknown but fixed Φ and Ψ . Define the observed region in cochain space: Cobs0:=⋃i=1Mx(i)([0,Ti])⊆C0(G;ℱ),Cobs1:=δ(Cobs0)⊆C1(G;ℱ). gatheredC^0_obs:= _i=1^Mx^(i)([0,T_i]) C^0(G;F),\\[5.0pt] C^1_obs:=δ(C^0_obs) C^1(G;F). gathered (10) Since Φ depends on y∈C1(G;ℱ)y∈ C^1(G;F) alone (Assumption 1), the residual (9) factors through δxδ x. We may therefore define the observable force map fobs(y):=δ∗Φ(y),y∈Cobs1.f_obs(y):=δ^* (y), y∈ C^1_obs. (11) Trajectories reveal only fobsf_obs, not Φ itself. The SI problem reduces to inverting the map Φ↦δ∗Φ δ^* on Cobs1C^1_obs. Lemma 2 (Identifiability obstruction). Two edge force laws Φ,Φ~:C1(G;ℱ)→C1(G;ℱ) , :C^1(G;F)→ C^1(G;F) satisfy δ∗Φ(y)=δ∗Φ~(y)δ^* (y)=δ^* (y) for all y∈Cobs1y∈ C^1_obs if and only if (Φ−Φ~)(y)∈kerδ∗( - )(y)∈ δ^* for all y∈Cobs1y∈ C^1_obs. By Lemma 1, kerδ∗≅H1(G;ℱ) δ^* H^1(G;F), so the dimension of the space of undetectable perturbations equals dimH1(G;ℱ) H^1(G;F).111For proofs and additional experimental details, see the Appendix. The following theorem characterizes when recovery is possible without parameterization. Theorem 2 (Unrestricted Recovery Boundary). Suppose the sheaf ℱF is fixed and known to the observer, and Ψ satisfies Assumption 2 with each WvW_v given as prior knowledge. Let ℳU:=∇U:U(y)=∑e∈EUe(ye),Ue∈C1(ℱ(e))M_U:= \∇ U:U(y)= _e∈ EU_e(y_e),\ U_e∈ C^1(F(e)) \ (12) be the set of conservative, edge-separable force maps. Then the following are equivalent: (i) For every Φ∈ℳU _U, no other Φ~∈ℳU _U satisfies δ∗Φ(y)=δ∗Φ~(y)δ^* (y)=δ^* (y) for all y∈Cobs1y∈ C^1_obs. (i) H1(G;ℱ)=0H^1(G;F)=0. When the obstruction vanishes, the recovered force law also determines each edge potential uniquely. Corollary 1. When the conditions of Theorem 2 hold, the recovered force Φ=∇U =∇ U uniquely determines each UeU_e on every path-connected component of πe(Cobs1) _e(C^1_obs), where πe _e is the projection onto ℱ(e)F(e), up to an additive constant. IV Parametric Potential Reconstruction When H1(G;ℱ)≠0H^1(G;F)≠ 0, Lemma 2 shows that kerδ∗≠0 δ^*≠ 0, so for any Φ∈ℳU _U the perturbation Φ+z +z with z∈kerδ∗z∈ δ^*, z≠0z≠ 0, is indistinguishable from Φ via trajectory data. Recovery from ℳUM_U is impossible; identifiability can be restored by restricting to a finite-dimensional parameterized class. For ease of exposition, we assume all edge stalks are isomorphic to a common space W≅ℝdW ^d, which holds when all communication channels employ the same comparison space—that is, when all edge stalks ℱ(e)F(e) are copies of the same space. The edge-heterogeneous case follows by applying the construction per isomorphism class. Fix a finite collection of smooth basis potentials ψ1,…,ψp _1,…, _p. With unknown coefficients θ=(θm)∈ℝpθ=( _m) ^p, define Uθ(y):=∑e∈E∑m=1pθmψm(ye).U_θ(y):= _e∈ E _m=1^p _m\, _m(y_e). (13) The θm _m are shared across all edges because we assume homogeneous stalks and a common interaction law: every agent pair uses the same potential family, differing only via the restriction maps encoded in δ. The corresponding force Φθ(y)=(∑mθm∇ψm(ye))e∈E _θ(y)=( _m _m∇ _m(y_e))_e∈ E is linear in θ. Theorem 3. Suppose we observe N distinct edge states y(1),…,y(N)∈Cobs1y^(1),…,y^(N)∈ C^1_obs with corresponding observed node signals fobs(i):=fobs(y(i))f^(i)_obs:=f_obs(y^(i)). (a) Linear parameterization: Define the design matrix A:ℝp→C0(G;ℱ)NA:R^p→ C^0(G;F)^N by (Aθ)i:=δ∗∇Uθ(y(i)),i=1,…,N.(Aθ)_i:=δ^*∇ U_θ(y^(i)), i=1,…,N. (14) The true parameter θ∗ _* is uniquely recoverable from the data fobs(i)\f^(i)_obs\ if and only if kerA=0 A=\0\, equivalently, the Gram matrix Γ:=A∗A :=A^*A is positive definite. (b) Nonlinear parameterization: For a nonlinear parameterization on open Θ⊆ℝp ^p, let N(θ):=(δ∗∇Uθ(y(i)))i=1N∈C0(G;ℱ)NO_N(θ):= (δ^*∇ U_θ(y^(i)) )_i=1^N∈ C^0(G;F)^N. The parameter θ∗ _* is locally identifiable if the Jacobian DN(θ∗)DO_N( _*) is injective, equivalently, the information matrix ℐN:=∑i=1NSi∗Si,Si:=Dθ[δ∗∇Uθ(y(i))]θ=θ∗,I_N:= _i=1^NS_i^*S_i, S_i:=D_θ[δ^*∇ U_θ(y^(i))]_θ= _*, (15) is positive definite. Remark 2. The edge states y(i)=δx(i)y^(i)=δ x^(i) entering Theorem 3 are computed from observed node states via the coboundary map δ; no direct measurement of edge quantities is required. Remark 3. Theorem 3 implies that identifiability depends jointly on: (i) the choice of basis functions ψm\ _m\, and (i) the diversity of edge states y(i)\y^(i)\ visited by the data—richer coverage of Cobs1C^1_obs leads to a better-conditioned ℐNI_N, while data confined to a low-dimensional region may leave ℐNI_N singular even when the basis is well chosen. For a fixed basis, one can optimize ℐNI_N via experimental design: trajectories should visit diverse regions of Cobs1C^1_obs, and basis functions should be chosen so that their images under δ∗δ^* span a large subspace of C0(G;ℱ)C^0(G;F). Figure 1: Experiment 1 on Sheaf B. The recovered edge law reaches the target formation; the rollout-only baseline does not. IV-A Estimation via Least Squares Given the parameterization (13), we observe discrete trajectory samples (xk,x˙k)k=1N\(x_k, x_k)\_k=1^N and compute residuals rk:=−x˙k−Ψ(xk)r_k:=- x_k- (x_k), which satisfy rk=δ∗∇Uθ(δxk)r_k=δ^*∇ U_θ(δ x_k) by (9). We estimate θ via θ^∈argminθ(1/N)∑k=1N‖rk−δ∗∇Uθ(δxk)‖C02+λℛ(θ), θ∈ _θ~(1/N)~ _k=1^N \|r_k-δ^*∇ U_θ(δ x_k) \|^2_C^0+ (θ), (16) where ℛR is a regularizer and λ≥0λ≥ 0. Parameterizing UθU_θ directly enforces conservativity by construction. When derivatives are noisy, one may instead minimize the integrated residual x(tk+1)−x(tk)≈−∫tktk+1[δ∗∇Uθ(δx(s))+Ψ(x(s))]ds,x(t_k+1)-x(t_k)≈- _t_k^t_k+1 [δ^*∇ U_θ(δ x(s))+ (x(s)) ]\,ds, (17) which avoids explicit differentiation and improves numerical stability. V Experiments Each experiment isolates a different obstruction to edge-law recovery—cohomological ambiguity, insufficient excitation, and finite-basis rank deficiency—and tests whether accurate node rollout implies accurate recovery of the local law. Throughout, Ψ=0 =0, so r(x)=−x˙r(x)=- x is directly observable. V-A Experiment 1: Cohomological Ambiguity in Formation Transfer We study sheaf dynamics on a directed cycle under a shifted quadratic edge potential (Example 1), whose equilibrium encodes a target agent formation. We construct a competing edge law that differs from the true law by an element of kerδ∗ δ^*—a perturbation invisible to node-level observations—and compare its effect on two sheaves: Sheaf A, for which H1(G;ℱ)≠0H^1(G;F)≠ 0, and Sheaf B, for which H1(G;ℱ)=0H^1(G;F)=0. On Sheaf A, the perturbed law produces identical node trajectories to the true law, so a rollout criterion accepts the wrong local rule and the system fails to reach the target formation; on Sheaf B, the same perturbation alters the rollout and is detectable. Figure 2 (left) illustrates both cases. Consequently, when H1(G;ℱ)≠0H^1(G;F)≠ 0, node-trajectory agreement does not certify edge-law recovery: distinct local laws can produce observationally equivalent dynamics. Figure 2: Nonparametric ambiguity and threshold recovery. V-B Experiment 2: Parametric Recovery under Bounded Confidence On a sheaf with H1(G;ℱ)=0H^1(G;F)=0, we recover the scalar confidence threshold ε of the bounded-confidence edge potential (Example 1), specialized to the smooth polynomial approximation Ue(ye;ε)=ψε(‖ye‖),ψε(r)=12r2−12ε2r4+16ε4r6,r≤ε,ε26,r>ε,U_e(y_e; )= _ (\|y_e\|), _ (r)= cases 12r^2- 12 ^2r^4+ 16 ^4r^6,&r≤ ,\\[6.00006pt] ^26,&r> , cases (18) under which each edge contributes a restoring force only when disagreement falls below ε . We vary data coverage (broad initial conditions spanning below, near, and above the threshold vs. localized conditions confined near the cutoff) and residual quality (directly observed vs. estimated by finite differences under node noise). Broad coverage recovers the threshold accurately under both residual regimes, while localized coverage degrades substantially when residuals are estimated from noisy data; in all cases, rollout error alone does not signal the failure. Figure 2 (right) illustrates these outcomes. The key lesson is that sufficient coverage of Cobs1C^1_obs—in particular, excitation of edge states near the threshold—is a necessary condition for recovery independent of any structural obstruction. V-C Experiment 3: Finite-Basis Identifiability We return to a sheaf with H1(G;ℱ)≠0H^1(G;F)≠ 0, where nonparametric recovery is impossible, and restrict the edge law to the finite-dimensional monomial basis class of Section IV. We test three conditions against the criterion of Theorem 3: the correct basis with broad data coverage; the correct basis augmented with a harmonic edge mode (a constant force that vanishes under δ∗δ^* and is therefore unidentifiable from trajectories); and the correct basis with limited initial-condition coverage. The correct basis with broad coverage yields a positive-definite information matrix Γ and accurate parameter recovery. The augmented basis renders Γ singular—the coefficient of the harmonic mode is not identifiable despite small rollout error—and limited coverage makes Γ nearly singular with similarly degraded recovery, again without a correspondingly large rollout error. Figure 3 confirms this picture. Identifiability in the parametric setting is thus governed by the rank of Γ , jointly determined by basis choice and data coverage, not by rollout accuracy. Figure 3: Finite-basis recovery and force-law error. VI Conclusion We studied when local interaction laws can be recovered from trajectory data in sheaf-based multi-agent systems. Our theory shows that recovery is determined by the sheaf cohomology of the system: when H1(G;ℱ)=0H^1(G;F)=0, the local law is uniquely recoverable from trajectory data; when H1(G;ℱ)≠0H^1(G;F)≠ 0, recovery requires both a restricted parametric class and sufficiently diverse data. In either regime, accurate node rollout does not certify recovery of the underlying local law—prediction and system identification are distinct objectives, and a learned model is interpretable only when the identifiability conditions are met. A natural next step is to develop estimation and experiment-design methods that remain reliable under noise, partial observation, and more general heterogeneous dynamics. A further limitation of the present work is that the node potential Ψ is assumed known and subtracted from the data as preprocessing; extending the identifiability theory to the joint recovery of both Φ and Ψ from trajectories is an important open problem. References -A Proofs This appendix proves Lemma 2, Theorem 2, its corollary, and Theorem 3. Throughout, gradients and adjoints are taken with respect to the inner products on the stalk and cochain spaces. Proof of Lemma 2. Fix y∈Cobs1y∈ C^1_obs. Then δ∗Φ(y)=δ∗Φ~(y) δ^* (y)=δ^* (y) (19) ⇔δ∗(Φ−Φ~)(y)=0 δ^*( - )(y)=0 ⇔(Φ−Φ~)(y)∈kerδ∗. ( - )(y)∈ δ^*. Since this holds for every y∈Cobs1y∈ C^1_obs, the first claim follows. By Lemma 1, kerδ∗≅H1(G;ℱ) δ^* H^1(G;F) as inner-product spaces. Hence, for each fixed observed edge state y, the vector space of undetectable perturbation values at y is exactly kerδ∗ δ^*, and therefore has dimension dimH1(G;ℱ) H^1(G;F). ∎ Proof of Theorem 2. We prove (i) ⇔ (i). (i) ⇒ (i). Assume H1(G;ℱ)=0H^1(G;F)=0. By Lemma 1, this is equivalent to kerδ∗=0 δ^*=\0\. If Φ,Φ~∈ℳU , _U satisfy δ∗Φ(y)=δ∗Φ~(y)for all y∈Cobs1,δ^* (y)=δ^* (y) all y∈ C^1_obs, (20) then Lemma 2 gives (Φ−Φ~)(y)∈kerδ∗=0for all y∈Cobs1.( - )(y)∈ δ^*=\0\ all y∈ C^1_obs. (21) Hence Φ(y)=Φ~(y) (y)= (y) for all y∈Cobs1y∈ C^1_obs, proving (i). (i) ⇒ (i). We prove the contrapositive. Suppose H1(G;ℱ)≠0H^1(G;F)≠ 0. By Lemma 1, choose 0≠z=(ze)e∈E∈kerδ∗0≠ z=(z_e)_e∈ E∈ δ^*. Fix any Φ=∇U∈ℳU =∇ U _U, where U(y)=∑e∈EUe(ye),Ue∈C1(ℱ(e)).U(y)= _e∈ EU_e(y_e), U_e∈ C^1(F(e)). (22) For each edge e, define ℓe(ξ) _e(ξ) :=⟨ze,ξ⟩e, = z_e,ξ _e, (23) U~e U_e :=Ue+ℓe, =U_e+ _e, U~(y) U(y) :=∑e∈EU~e(ye). = _e∈ E U_e(y_e). Then each U~e U_e is C1C^1, so Φ~:=∇U~ :=∇ U belongs to ℳUM_U. Since ∇ℓe=ze∇ _e=z_e, we have Φ~(y)=∇U~(y)=∇U(y)+z=Φ(y)+z (y)=∇ U(y)=∇ U(y)+z= (y)+z (24) for all y∈C1(G;ℱ)y∈ C^1(G;F). Thus Φ~≠Φ ≠ . But for every y∈Cobs1y∈ C^1_obs, δ∗Φ~(y)=δ∗(Φ(y)+z)=δ∗Φ(y)+δ∗z=δ∗Φ(y),δ^* (y)=δ^*( (y)+z)=δ^* (y)+δ^*z=δ^* (y), (25) because z∈kerδ∗z∈ δ^*. Hence (i) fails. This proves the contrapositive. ∎ Proof of the Corollary. Under Theorem 2, the force law Φ=∇U =∇ U is uniquely determined on Cobs1C^1_obs. Fix e∈Ee∈ E and let Se:=πe(Cobs1)⊆ℱ(e).S_e:= _e(C^1_obs) (e). (26) Because U is edge-separable, Φe(y)=∇Ue(ye)for all y∈C1(G;ℱ). _e(y)=∇ U_e(y_e) all y∈ C^1(G;F). (27) Hence if y,y¯∈Cobs1y, y∈ C^1_obs satisfy πe(y)=πe(y¯) _e(y)= _e( y), then Φe(y)=∇Ue(ye)=∇Ue(y¯e)=Φe(y¯). _e(y)=∇ U_e(y_e)=∇ U_e( y_e)= _e( y). (28) Therefore ge:Se→ℱ(e),ge(ξ):=Φe(y),g_e:S_e (e), g_e(ξ):= _e(y), (29) for any y∈Cobs1y∈ C^1_obs with πe(y)=ξ _e(y)=ξ, is well defined, and ge=∇Ueg_e=∇ U_e on SeS_e. Let Q be a piecewise-C1C^1 path-connected component of SeS_e, and fix ξ0∈Q _0∈ Q. For any ξ∈Qξ∈ Q, choose a piecewise-C1C^1 path γ:[0,1]→Qγ:[0,1]→ Q with γ(0)=ξ0γ(0)= _0 and γ(1)=ξγ(1)=ξ. Along each C1C^1 segment of γ, the chain rule gives dtUe(γ(t))=⟨∇Ue(γ(t)),γ˙(t)⟩e=⟨ge(γ(t)),γ˙(t)⟩e. \,d\,dtU_e(γ(t))= ∇ U_e(γ(t)), γ(t) _e= g_e(γ(t)), γ(t) _e. (30) Integrating along the segments yields Ue(ξ)−Ue(ξ0)=∫01⟨ge(γ(t)),γ˙(t)⟩edt.U_e(ξ)-U_e( _0)= _0^1 g_e(γ(t)), γ(t) _e\,\,dt. (31) Thus geg_e determines UeU_e on Q up to the additive constant Ue(ξ0)U_e( _0). If U^e∈C1(ℱ(e)) U_e∈ C^1(F(e)) also satisfies ∇U^e=ge∇ U_e=g_e on Q, then h:=Ue−U^eh:=U_e- U_e satisfies ∇h=0∇ h=0 on Q. For any piecewise-C1C^1 path γ in Q from ξ0 _0 to ξ, h(ξ)−h(ξ0)=∫01⟨∇h(γ(t)),γ˙(t)⟩edt=0.h(ξ)-h( _0)= _0^1 ∇ h(γ(t)), γ(t) _e\,\,dt=0. (32) Hence h is constant on Q, so U^e U_e differs from UeU_e by an additive constant on Q. ∎ Proof of Theorem 3. We treat (a) and (b) separately. (a) Linear parameterization. For each i∈1,…,Ni∈\1,…,N\ and m∈1,…,pm∈\1,…,p\, set ζm(i):=(∇ψm(ye(i)))e∈E∈C1(G;ℱ). _m^(i):= (∇ _m(y_e^(i)) )_e∈ E∈ C^1(G;F). (33) Then ∇Uθ(y(i)) ∇ U_θ(y^(i)) =∑m=1pθmζm(i), = _m=1^p _m\, _m^(i), (34) δ∗∇Uθ(y(i)) δ^*∇ U_θ(y^(i)) =∑m=1pθmδ∗ζm(i). = _m=1^p _m\,δ^* _m^(i). Hence the map A:ℝp→C0(G;ℱ)NA:R^p→ C^0(G;F)^N defined by Aθ:=(δ∗∇Uθ(y(1)),…,δ∗∇Uθ(y(N))),Aθ:= (δ^*∇ U_θ(y^(1)),…,δ^*∇ U_θ(y^(N)) ), (35) is linear. In the noiseless setting, Aθ∗=(fobs(1),…,fobs(N)).A _*=(f_obs^(1),…,f_obs^(N)). (36) Thus θ∗ _* is uniquely determined by the data if and only if Aθ=Aθ∗Aθ=A _* implies θ=θ∗θ= _*, i.e., if and only if kerA=0 A=\0\. Equip C0(G;ℱ)NC^0(G;F)^N with the product inner product and ℝpR^p with its standard Euclidean inner product. Then, for every v∈ℝpv ^p, ⟨v,Γv⟩=⟨v,A∗Av⟩=⟨Av,Av⟩=‖Av‖2. v, v = v,A^*Av = Av,Av =\|Av\|^2. (37) Therefore Γ=A∗A =A^*A is positive definite if and only if Av≠0Av≠ 0 for every v≠0v≠ 0, equivalently, if and only if kerA=0 A=\0\. (b) Nonlinear parameterization. Let J:=DN(θ∗):ℝp→C0(G;ℱ)N.J:=DO_N( _*):R^p→ C^0(G;F)^N. (38) Assume J is injective. Then J is an isomorphism from ℝpR^p onto the p-dimensional subspace imJ⊆C0(G;ℱ)NimJ C^0(G;F)^N, so there exists a linear map P:C0(G;ℱ)N→ℝpP:C^0(G;F)^N ^p (39) such that PJ=IℝpPJ=I_R^p. Define F:=P∘NF:=P _N. Then F is C1C^1 and DF(θ∗)=PDN(θ∗)=PJ=Iℝp.DF( _*)=P\,DO_N( _*)=PJ=I_R^p. (40) By the inverse function theorem, F is locally one-to-one near θ∗ _*. Hence, if θ1,θ2 _1, _2 are sufficiently close to θ∗ _* and N(θ1)=N(θ2)O_N( _1)=O_N( _2), then F(θ1)=PN(θ1)=PN(θ2)=F(θ2),F( _1)=PO_N( _1)=PO_N( _2)=F( _2), (41) so θ1=θ2 _1= _2. Thus NO_N is locally injective at θ∗ _*, and θ∗ _* is locally identifiable. Finally, by definition of SiS_i, Jv=(S1v,…,SNv)for all v∈ℝp.Jv=(S_1v,…,S_Nv) all v ^p. (42) Therefore ‖Jv‖2 \|Jv\|^2 =∑i=1N‖Siv‖C02 = _i=1^N\|S_iv\|_C^0^2 (43) =∑i=1N⟨v,Si∗Siv⟩ = _i=1^N v,S_i^*S_iv =⟨v,(∑i=1NSi∗Si)v⟩ = v, ( _i=1^NS_i^*S_i )v =⟨v,ℐNv⟩. = v,I_Nv . Hence ℐNI_N is positive definite if and only if J is injective, which is the claimed equivalence. ∎ -B Additional experimental details All node and edge stalks are two-dimensional; trajectories are integrated with step size 0.010.01. Rollout errors compare node trajectories; force-law errors compare the recovered interaction in edge space. Results for Experiments 2 and 3 are reported as mean ± standard deviation over eight independent seeds, except the deterministic augmented-basis row in Table I. Experiment 1: Setup Details The experiment uses three agents on a directed 33-cycle. The shifted-quadratic edge potential is Ue(ye;be)=12‖ye−be‖2,∇Ue(ye;be)=ye−be.U_e(y_e;b_e)= 12\|y_e-b_e\|^2, ∇ U_e(y_e;b_e)=y_e-b_e. (44) The true system is simulated for 44 seconds; its terminal state defines the target formation. Agents are then reset and coordinated with two candidate laws: the recovered-edge-law model, and a rollout-only alternative Φ~(y)=y+βc (y)=y+β c with β=0.6β=0.6 and c=(1,0)c=(1,0) on each edge. Sheaf A uses identity tail maps (dimH1(G;ℱ)=2 H^1(G;F)=2); Sheaf B uses rotated tail maps (H1(G;ℱ)=0H^1(G;F)=0). On Sheaf A, the perturbation βc∈kerδ∗β c∈ δ^* is invisible to trajectory observations; on Sheaf B, the same perturbation alters the rollout and the terminal formation. Numerical results are reported in Table I. Experiment 2: Setup Details We use the rotated 33-cycle sheaf (H1(G;ℱ)=0H^1(G;F)=0) and set the true threshold ε⋆=1 _ =1. Broad sampling uses initial-condition scales (0.4,0.8,1.2)(0.4,0.8,1.2), placing edge states below, near, and above the threshold; localized sampling uses only a thin annulus near the cutoff. Residuals are either observed directly or estimated by finite differences after adding Gaussian node noise with standard deviation 5×10−35× 10^-3. Table I reports recovery error, rollout RMSE, and information number ℐNI_N across conditions. Force-law MSE on pooled and grid evaluation sets is reported in Table IV. On the 55-cycle under the same sheaf, broad sampling with observed residuals gives |ε^−ε⋆|=2.21×10−9| - _ |=2.21× 10^-9, while localized sampling with finite-difference residuals gives 2.54×10−22.54× 10^-2. Experiment 3: Setup Details We use the identity 33-cycle sheaf (dimH1(G;ℱ)=2 H^1(G;F)=2) with the monomial edge law Φθ(ye)=ye(θ1+θ2‖ye‖2+θ3‖ye‖4),θ⋆=(1, 0.25, 0.03). _θ(y_e)=y_e ( _1+ _2\|y_e\|^2+ _3\|y_e\|^4 ), _ =(1,\,0.25,\,0.03). (45) For finite-difference residuals, Gaussian node noise with standard deviation 10−410^-4 is added. The augmented basis appends a constant force c∈kerδ∗c∈ δ^* that is invisible to δ∗δ^* and therefore produces a singular Γ . Table I reports relative parameter error, rollout RMSE, and λmin(Γ) _ ( ) across conditions. Table IV shows that models with limited coverage can match holdout trajectories while failing on pooled observed edge states and a reference grid. On the identity 55-cycle, the relative parameter errors for the correct/broad/Obs., augmented/Obs., and correct/limited/Obs. conditions are 1.15×10−121.15× 10^-12, 4.36×10−14.36× 10^-1, and 1.18×10−11.18× 10^-1, respectively. TABLE I: Nonparametric ambiguity on cycle sheaves. Rollout difference measures node trajectories; force MSE measures edge laws. Sheaf dimH1 H^1 Max rollout diff. Force MSE 3-cycle, Sheaf A 2 7.73×10−167.73× 10^-16 1.081.08 3-cycle, Sheaf B 0 7.60×10−17.60× 10^-1 1.081.08 5-cycle, Sheaf A 2 7.47×10−167.47× 10^-16 1.801.80 5-cycle, Sheaf B 0 8.07×10−18.07× 10^-1 1.801.80 TABLE I: Recovery of the bounded-confidence threshold on the rotated 33-cycle sheaf. Obs. = observed residuals; FD = finite differences. Entries are mean ± standard deviation over eight seeds. Setting |ε^−1|| -1| Rollout RMSE ℐNI_N Broad / Obs. (2.43±1.05)×10−9(2.43± 1.05)× 10^-9 (1.43±1.22)×10−9(1.43± 1.22)× 10^-9 (2.03±0.30)×103(2.03± 0.30)× 10^3 Localized / Obs. (5.74±3.07)×10−9(5.74± 3.07)× 10^-9 (9.67±5.13)×10−10(9.67± 5.13)× 10^-10 (2.08±0.09)×102(2.08± 0.09)× 10^2 Broad / FD (1.08±0.54)×10−3(1.08± 0.54)× 10^-3 (6.16±5.81)×10−4(6.16± 5.81)× 10^-4 (1.99±0.29)×103(1.99± 0.29)× 10^3 Localized / FD (1.88±3.12)×10−1(1.88± 3.12)× 10^-1 (8.56±1.98)×10−4(8.56± 1.98)× 10^-4 (2.86±0.09)×102(2.86± 0.09)× 10^2 TABLE I: Recovery in the finite basis class on the identity 33-cycle sheaf. Obs. = observed residuals; FD = finite differences. Entries are mean ± standard deviation over eight seeds, except the deterministic augmented-basis row. Setting Rel. param. err. Rollout RMSE λmin(Γ) _ ( ) Correct / Broad / Obs. (2.07±0.32)×10−12(2.07± 0.32)× 10^-12 (1.16±0.13)×10−13(1.16± 0.13)× 10^-13 (4.12±0.79)×103(4.12± 0.79)× 10^3 Augmented / Obs. 4.36×10−14.36× 10^-1 1.15×10−131.15× 10^-13 0 Correct / Limited / Obs. (1.67±0.20)×10−1(1.67± 0.20)× 10^-1 (7.68±1.03)×10−9(7.68± 1.03)× 10^-9 (9.29±6.67)×10−17(9.29± 6.67)× 10^-17 Correct / Broad / FD (1.20±0.71)×10−1(1.20± 0.71)× 10^-1 (3.06±1.56)×10−3(3.06± 1.56)× 10^-3 (2.33±0.30)×103(2.33± 0.30)× 10^3 Correct / Limited / FD 8.41±5.798.41± 5.79 (2.23±1.74)×10−6(2.23± 1.74)× 10^-6 (6.84±5.26)×10−17(6.84± 5.26)× 10^-17 TABLE IV: Median force-law MSE on three evaluation sets for the 33-cycle. The pooled set and reference grid expose errors hidden by holdout trajectory error. Setting Holdout Pooled Grid Bounded-confidence threshold recovery Broad / Obs. 3.99×10−193.99× 10^-19 6.78×10−196.78× 10^-19 3.67×10−193.67× 10^-19 Localized / Obs. 1.21×10−181.21× 10^-18 6.69×10−186.69× 10^-18 3.62×10−183.62× 10^-18 Broad / FD 6.55×10−86.55× 10^-8 1.85×10−71.85× 10^-7 1.00×10−71.00× 10^-7 Localized / FD 1.30×10−61.30× 10^-6 9.41×10−59.41× 10^-5 5.25×10−55.25× 10^-5 Finite basis class Correct / Broad / Obs. 6.97×10−256.97× 10^-25 1.03×10−241.03× 10^-24 1.34×10−241.34× 10^-24 Augmented / Obs. 7.50×10−17.50× 10^-1 7.50×10−17.50× 10^-1 7.50×10−17.50× 10^-1 Correct / Limited / Obs. 4.15×10−154.15× 10^-15 6.53×1006.53× 10^0 5.26×1005.26× 10^0 Correct / Broad / FD 1.08×1001.08× 10^0 8.76×10−18.76× 10^-1 4.91×10−24.91× 10^-2 Correct / Limited / FD 1.25×10−101.25× 10^-10 9.78×1029.78× 10^2 3.19×1033.19× 10^3