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Learning Interaction Kernels from Collective Steady States
Baoli Hao, Mauro Maggioni, Ming Zhong
TL;DR
Recovering interaction laws from collective steady states is ill-posed because temporal information is absent and different kernels can yield similar patterns. The paper recasts recovery as a weighted variational eigenvalue problem and finds that identifiability varies by observed regime.
Problem
Without temporal information, different interaction mechanisms can induce almost indistinguishable collective steady-state patterns, making kernel recovery ill-posed.
Source: §1 · p. 2Method
A ρT-weighted variational problem recasts recovery as an eigenvalue problem to estimate the kernel on the support of the observed pairwise-distance distribution.
Source: §5 · p. 33Results
Identifiability depends on regime: nonzero-right-hand-side moving flocks and milling are well-posed, while static states and second-order flocking identify kernels only up to a scale.
Source: §5 · p. 33Takeaways & Limitations
Collective steady-state patterns can constrain the laws that produced them when interpreted through the appropriate structure.
Source: §5 · p. 34Recovery is constrained to supp(ρT), and open questions include noise or partial observation and fixing the missing scale using auxiliary information.
Source: §5 · p. 34
Abstract
from arXiv · showhide
We propose a learning procedure for system identification in interacting particle systems from single-snapshot observations of collective behaviors, unlike existing approaches that rely on observations of trajectories. This setting leads to a fundamentally ill-posed inverse problem, which we solve by using a regularization strategy based on the empirical distribution of observed configurations, drawn from different, unobserved initial conditions. We test our learning procedure on a variety of representative models with steady-state and quasi-stationary patterns, where collective behaviors encode implicit information about the interaction mechanisms, demonstrating that our approach enables stable and accurate recovery of the underlying interaction laws, leading to faithful reproduction of the collective behavior, and in many cases even of the dynamics leading up to it.
1. Introduction
The paper addresses recovery of interaction laws from collective steady states, where temporal information is absent and distinct mechanisms can produce nearly indistinguishable patterns. It combines variational learning with eigenvalue solvers and tests recovery across several steady and quasi-steady regimes.
1. Introduction
- Without trajectory data, recovering interaction laws from collective steady states is ill-posed because distinct mechanisms can produce nearly indistinguishable patterns.Existing approaches typically use trajectories to obtain information about time derivatives.Source: §1 · p. 2
- The particle configuration at a collective steady state contains enough structure to estimate interaction kernels, and experiments report robust recovery under increased observation noise.The experiments span first-order steady states, multi-species semi-steady states, and fish-milling quasi-steady states.Source: §1 · p. 2
1.1. Comparison to Other Methods
- Prior interacting-particle methods typically infer kernels from trajectory data using variational inverse problems and symmetry-aware bases.The passage also notes related approaches using Gaussian Processes, Random Features, and mean-field limits.Source: §1.1 · p. 3
- The learning method combines a variational approach with eigenvalue solvers to recover interaction kernels with limited time information.The method recasts learning as an eigenvalue problem.Source: §1.1 · p. 3
2. Model Equations
The paper considers first- and second-order interacting-particle models, with kernels acting on low-dimensional features of agent states. It assumes known agent types and dynamics that approach stationary distributions or collective steady states.
2. Model Equations
- In first-order models, agents’ state changes are governed by ODEs, with pairwise interactions represented by kernels acting on relative positions.The models can include an external, non-collective force.Source: §2 · pp. 3–4
- The models allow heterogeneous agents partitioned into types, and each agent’s type is assumed known.There are interaction kernels for each pair of types.Source: §2 · pp. 3–4
- Second-order models use potential-energy-based kernels for interactions and separate kernels to model alignment.The models may also include an environment-dependent, non-collective force.Source: §2 · p. 4
- The interaction kernels and non-collective forces are assumed to satisfy regularity conditions ensuring solutions exist uniquely for all time.Source: §2 · p. 4
- With no external force, a single-species first-order system is a gradient flow that evolves toward a minimum-energy state.Collective-system energies often have translation, rotation, and permutation invariance, leading to radial dependence.Source: §2 · p. 4
- Kernels may depend on a reduced feature map, commonly pairwise distance, lowering the feature dimension and the number of parameters to estimate.The feature map is assumed known, and its output dimension is much smaller than the full pairwise state dimension.Source: §2 · p. 4
- Initial states are drawn from distributions, and the models are assumed to approach stationary distributions or attractive collective steady states by a sufficiently large time.This includes behaviors such as flocking and milling.Source: §2 · p. 5
3. Learning Framework
The framework learns interaction kernels from independent single-time snapshots of collective steady states and characterizes the resulting static learning problem through identifiability, shared radial self-stresses, and stability. It addresses ill-conditioning and scaling ambiguities using minimal-energy information, stopping times, and data-driven partitions.
3. Learning Framework
- The framework fits interaction kernels by minimizing empirical losses over finite-dimensional hypothesis spaces using independent, identically distributed single-time snapshots from the stationary collective-state distribution.The snapshots may come from trajectories begun at independently sampled initial conditions.Source: §3 · p. 5
3.1. Performance Measures
- Kernel accuracy is measured in pair-distance-distribution-weighted L2 norms, while emergent-pattern reproduction is assessed separately using trajectory error or regime-specific flocking and milling scores.The weighted norms use pairwise-distance distributions for each interaction type; flocking and milling scores replace trajectory error in semi-static and quasi-static regimes.Source: §3.1 · pp. 6–7
3.2. Ill-conditioned estimation of the interaction kernels
- Collective steady-state losses can be ill-conditioned or ill-posed; second-order flocking erases alignment-kernel information, motivating a weighted-norm regularity constraint on equilibrium-residual minimization.For second-order flocking, identical velocities make the alignment term vanish; the constrained loss is normalized by the weighted kernel norm.Source: §3.2 · pp. 7–9
3.3. Identifiability
- Under first-order steady states or second-order flocking, a one-dimensional equilibrium nullspace identifies the kernel direction, but snapshots alone leave its sign and scale undetermined.The guarantee is in a weighted L2 norm and is uninformative when observed pairwise distances concentrate at a few points or outside the kernel support, as can occur under attraction.Source: §3.3 · pp. 9–11
3.4. Finite-dimensional spectral characterization
- A simple smallest generalized eigenvalue yields directional stability governed by the spectral gap: one-dimensional nullity gives exact direction, large gaps stabilize estimates, and small gaps make alternatives hard to distinguish.Small gaps signal that other kernel directions produce nearly the same equilibrium residual, increasing difficulty under sampling error, noise, or model mismatch.Source: §3.4 · p. 11
- Static kernel learning identifies shared radial self-stresses; directional identifiability requires a one-dimensional nullspace, and depends on configurations and the hypothesis space.Geometric symmetries can reduce the number of independent equilibrium constraints.Source: §3.4 · p. 12
3.5. Minimal Energy Direction
- Steady-state data identifies the estimator only up to independent block-scaling signs; for gradient-flow systems, minimal-energy information can resolve those signs.The method uses the fact that a gradient-flow steady state also minimizes energy, evaluating perturbed configurations to select the sign matrix.Source: §3.5 · p. 12
3.6. Time-Scale Recovery
- Initial conditions and stopping times can recover the remaining scaling: in single-species systems stopping times give it in closed form, while multispecies systems match simulated and observed stopping times.Source: §3.6 · p. 13
3.7. Data-driven partitions
- Adaptive partitions concentrate basis functions where empirical distance mass lies, limit each cell’s mass, and improve conditioning by removing weakly observed directions.The resulting partition is approximately equidistributed under the empirical mass.Source: §3.7 · p. 14
4. Examples
The examples test kernel recovery and dynamical prediction across static, semi-static, and quasi-static collective patterns. Evaluation considers both agreement with the observed terminal pattern and prediction from new initial configurations.
4. Examples
- The examples span static first-order patterns, semi-static flocking in first- and second-order systems, and quasi-static milling in second-order self-propelled systems.Static examples include rings, crystals, soccer-ball configurations, and predator–prey aggregates.Source: §4 · p. 14
- Evaluation separates kernel-level comparison, including a scale-ambiguous zero-right-hand-side regime, from tests of terminal-pattern preservation and prediction from new initial configurations.The kernel comparison uses the weighted norm when the right-hand side is nonzero; dynamical tests assess both the observed terminal pattern and emergent patterns from new starts.Source: §4 · pp. 14–15
- All experiments use simulated training data, with common testing parameters summarized in Table 1 and M denoting the number of training initial conditions.Mtrue is the larger reference ensemble used to approximate the background measure.Source: §4 · p. 15, Table 1 · p. 15
- Pattern-specific stopping criteria and order scores are used across the examples, with tolerances set by regime and noise ranges chosen per example.The stated tolerances are 10^-9 for static and semi-static regimes and 10^-6 for quasi-static milling.Source: §4 · p. 15
4.1. Static Patterns from First-Order Systems
For static first-order systems, recovery is governed by the geometry of the equilibrium pairwise-distance distribution ρT: the interaction kernel is recoverable only on its support. The examples progress from finite support through clustered support to a continuous interval, testing how broader coverage changes recovery.
4.1.1. Ring Pattern
- For the ring, learned ϕ matches exactly on supp(ρT), and rescaled learned dynamics track true trajectories from fresh starts to convergence.Against interpolation from observed values, the variational estimate stays close across tested N, while interpolation becomes unstable between sparse points.Source: §4.1.1 · pp. 18–19
4.1.2. Crystal Formation
- Crystal recovery has θ = 9.81 × 10−4; rescaled dynamics reproduce fresh-start trajectories, but clustered ρT leaves gaps, especially in the poorly sampled repulsive core.As N grows, clusters spread and both reconstructions converge to ϕ on supp(ρT), but gaps remain a coverage limit.Source: §4.1.2 · p. 20
4.1.3. Soccer Ball Pattern
- For the soccer-ball pattern, continuous ρT coverage lets the learned kernel track ϕ across its support, while fresh-start trajectories are reproduced across tested N.Coverage is already adequate at small N, and the interpolation baseline is also accurate across tested N.Source: §4.1.3 · p. 21
4.1. Static Patterns from First-Order Systems
- Across homogeneous patterns, ρT-adaptive recovery stays accurate on observed support; expanding support from points to clusters to a continuous interval narrows what remains unlearnable.The study orders these cases from most to least ill-conditioned steady-state data.Source: §4.1 · p. 16, §4.1.3 · p. 21
4.1.4. Predator–Prey Model: Static Patterns
- In predator–prey recovery, all four kernels reproduce fresh-start trajectories; predator-directed kernels are least accurate at Np = 12, converging on their supports as N grows.The prey-directed kernels are near-exact, and the predator-related kernels are constrained by fewer equations.Source: §4.1.4 · p. 23
4.2. Semi-static Patterns
Semi-static patterns include moving flocks whose identifiability depends on whether the learning equation’s right-hand side vanishes. Across first- and second-order examples, learned interactions reproduce observed flocking and often trajectories from new initial conditions.
4.2. Semi-static Patterns
- For the semi-static regime, first-order moving flocks have nonzero limiting velocity, whereas second-order flocks have vanishing acceleration and require angular-error and scale recovery.Flocking score evaluates learned dynamics because the pattern is velocity alignment rather than convergence to a fixed spatial configuration.Source: §4.2 · pp. 23–24
4.2.1. First-order Predator–Prey Model: Flocking
- First-order predator–prey flocking is well-conditioned because its nonzero learning-equation right-hand side identifies kernels directly without scaling correction.This contrasts with the static predator–prey case and the second-order flocking examples, where the right-hand side vanishes and scaling ambiguity returns.Source: §4.2 · p. 23, §4.2.1 · p. 25
- The learned predator–prey kernels accurately recover the four directed interactions on observed distance supports and reproduce the flock from new initial conditions.The learned dynamics also preserve the observed terminal flock and produce a coherent flocking score.Source: Figure 10 · p. 24, §4.2.1 · p. 25, Figure 11 · p. 25
4.2.2. Second-order Fish Flocking
- In second-order fish flocking, vanishing acceleration and self-propulsion force make the learning equation’s right-hand side zero, leaving the energy kernel identifiable only up to scale.The positive scaling is recovered using the first-stopping-time procedure.Source: §4.2.2 · pp. 26–27
- In second-order fish flocking, the rescaled learned energy kernel preserves the observed flock and reproduces trajectories from new initial conditions.The narrow observed distance support means larger agent counts stabilize recovery, while small counts leave gaps.Source: §4.2.2 · pp. 26–27, Figure 12 · p. 26, Figure 13 · p. 27
4.2.3. Second-order Flocking with Anticipation Dynamics
- Flocking data observe only the ξ = 0 slice of the anticipated energy kernel, so its ξ-dependence comes from model structure rather than observations.The one-dimensional alignment kernel and potential agree with truth on the observed distance support.Source: §4.2.3 · p. 28, Figure 14 · p. 29
- Anticipation flocking data cannot identify the alignment kernel directly, but the shared-potential structure permits its reconstruction alongside the energy kernel.The reconstructed interactions preserve the observed flock and reproduce trajectories from fresh initial conditions.Source: §4.2.3 · pp. 28–29
4.3. Quasi-static Patterns from Second-Order Systems
Quasi-static milling produces persistent rotation with frozen pairwise distances, and nonzero centripetal acceleration makes kernel recovery well-posed without scale recovery. The double mill shows that kernel error and fidelity to collective geometry can diverge.
4.3. Quasi-static Patterns from Second-Order Systems
- Milling remains identifiable without scaling because its centripetal acceleration is nonzero, unlike second-order flocking where acceleration vanishes.The examples vary the repulsion parameter to produce single-ring and double-mill patterns.Source: §4.3 · p. 30, §4.3.1 · p. 31
4.3.1. Fish Milling: Rings
- For single-ring milling, rigid rotation allows centripetal acceleration to be reconstructed from positions, center, and speed despite acceleration not being directly observed.The reconstruction applies to clockwise and counterclockwise subgroups and yields direct weighted kernel errors without scale recovery.Source: §4.3.1 · p. 31
- Single-ring milling yields a small weighted kernel error and near-unit milling score, with accurate recovery on the narrow band of observed pairwise distances.The rigid-rotation reconstruction assumes all agents orbit one common center.Source: Figure 16 · p. 30, §4.3.1 · p. 32
4.3.2. Fish Milling: Double Milling
- The double mill’s opposed circulations invalidate the single-rotation acceleration reconstruction, so accelerations are used as observed data.Recovering local rotational structure from positions and velocities is outside the scope of this kernel-learning analysis.Source: §4.3.2 · p. 32, Figure 18 · p. 32
- The double-mill kernel error is three orders of magnitude larger than the single-ring error, yet the learned flow preserves the double mill and has near-unit milling score.The broader, multimodal distance distribution contributes to the larger error; the scalar milling score alone cannot distinguish one ring from two counter-rotating groups.Source: Table 7 · p. 31, §4.3.2 · p. 32, §4.3.1 · p. 32
- The learned dynamics recover double-mill type with over 99% accuracy, persistence within 10^-3 on average, and subgroup count in roughly 81% of realizations.The count is the strictest discrete diagnostic and is more sensitive to borderline assignments.Source: §4.3.2 · p. 33, Table 8 · p. 33, Table 9 · p. 33
5. Conclusion
The paper formulates single-snapshot interaction recovery as a ρT-weighted variational eigenvalue problem, with identifiability depending on the observed regime and distance support. Examples show that structural constraints can supply information missing from data, while kernel accuracy alone does not guarantee pattern fidelity.
5. Conclusion
- A ρT-weighted variational formulation turns recovery into an eigenvalue problem that estimates the kernel on observed distance support, assuming suitable identifiability.Single-snapshot data can be degenerate when temporal information is absent or the learning-equation right-hand side vanishes.Source: §5 · p. 33
- A nonzero learning-equation right-hand side yields well-posed recovery, while a vanishing one leaves scale ambiguity; observed distance support limits what is identifiable.A shared potential can recover an alignment kernel unavailable from flocking data, and rigid-rotation constraints can supply unobserved accelerations.Source: §5 · pp. 33–34
- A larger kernel error need not destroy emergent geometry, so recovery should also be assessed through the collective pattern it generates.The double-mill example motivates diagnostics beyond kernel error alone.Source: §5 · p. 34
- Open questions include recovery under noise or partial observation and how distance-distribution geometry controls the observation operator’s null space and spectral gap.The paper also identifies recovering missing scale from auxiliary information as an open direction.Source: §5 · p. 34
Appendix A. Numerical Implementation
For scale-ambiguous static and second-order flocking data, the implementation recovers a normalized kernel direction through constrained minimization, expressed as a generalized eigenvalue problem. Data-adaptive basis directions unsupported by observed distances are removed, and stability indicators are interpreted within each problem.
Appendix A. Numerical Implementation
- The adaptive basis retains directions observable under ρT; directions in the Gram matrix’s null space are removed before solving.The implementation assumes the Gram matrix is positive definite on the retained coefficient space.Source: Appendix A · p. 35
- The homogeneous residual minimization for scale-ambiguous data is solved as a symmetric generalized eigenvalue problem, with the smallest-eigenvalue eigenvector providing the normalized kernel estimator.The interaction equations decouple by receiving type, so one eigenvalue problem is solved for each type.Source: Appendix A · pp. 34–36
- The generalized spectral gap measures separation from the next competing kernel direction, and a positive gap gives uniqueness up to sign with perturbation sensitivity controlled by a proposition.Eigenvalues and gaps are within-problem stability indicators, not directly comparable condition numbers across different examples or hypothesis spaces.Source: Appendix A · p. 36
- The eigenvector supplies only a normalized kernel direction; the physical sign comes from a minimal-energy criterion and remaining positive scaling from stopping times.For a homogeneous system, the construction reduces to one generalized eigenvalue problem.Source: Appendix A · p. 36
Appendix B. Additional Examples
Appendix B adds learning examples across several pattern types and examines comparisons with B-spline interpolation and the effects of noise.
Appendix B. Additional Examples
- Figures B.19–B.20 compare learning ϕ at N ∈ {10, 20, 40, 80} with B-spline interpolation ˆϕinterp.The captions also identify ρM in the background.Source: Figure B.19 · p. 37, Figure B.20 · p. 38
- Figure B.21 presents static predator–prey learning of ϕk1,k2 across three (N, Ns, Np) settings, compared with B-spline interpolation ˆϕinterp.The settings are (10, 7, 3), (20, 14, 6), and (40, 28, 12).Source: Figure B.21 · p. 38
- Figure B.22 presents flocking predator–prey learning of ϕk1,k2 across the same three (N, Ns, Np) settings, compared with B-spline interpolation ˆϕinterp.The settings are (10, 7, 3), (20, 14, 6), and (40, 28, 12).Source: Figure B.22 · p. 39
- The cited figure captions report additional learning from steady patterns in four settings: LJ-crystal (Figure B.23), anticipated flocking (B.25), soccer ball (B.26), and fish flocking (B.27).Source: Figure B.23 · p. 39, Figure B.25 · p. 40, Figure B.26 · p. 40
- Noise effects in learning are shown for flocking predator–prey and quasi-static patterns from second-order systems.These are Figures B.24 and B.28, respectively.Source: Figure B.24 · p. 39, Figure B.28 · p. 41