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Schrödinger Bridges over Kinetic Swarming Models

Asmaa Eldesoukey, Md Zulfiqur Haider, Italo Napolitano, Yongxin Chen, Abhishek Halder

arXiv:2608.25281v1math.OCeess.SYmath-phstat.ML

TL;DR

The paper addresses finite-horizon minimum-energy steering of noisy interacting inertial swarms when interactions alone may not reach prescribed aggregate configurations. It extends Schrödinger bridges to mean-field Cucker–Smale and Morse kinetics with full phase-space or position-marginal endpoints, deriving computable optimality systems. Numerically, the control exploits favorable interactions and counteracts adverse ones, with lower costs than interaction-canceling or classical non-interacting baselines in the reported examples.

  • Problem

    Interactions may not bring noisy swarms to desired aggregate configurations within a prescribed time, motivating collective steering between endpoint distributions.

  • Method

    The paper formulates Schrödinger-bridge minimum-energy control for mean-field Cucker–Smale and Morse kinetic models with phase-space or position-marginal endpoint constraints.

  • Results

    The computed control adapts to interactions, exploiting favorable forces or counteracting adverse ones, and has lower reported cost than the compared baselines.

  • Takeaways & Limitations

    Interacting kinetic dynamics can be incorporated into Schrödinger-bridge steering while allowing the corrective drift to adapt its effort to the interaction forces.

  • Takeaways & Limitations

    The nonconvex formulations yield first-order necessary conditions, so computed solutions are stationary points rather than guaranteed global minima.

Abstract

from arXiv · show

Paradigmatic interaction models explain how collective behaviors can emerge in complex systems from interactions among the constituent agents. In bio-inspired swarms, however, interactions alone may not suffice to bring the population to a desired aggregate configuration within a prescribed time horizon, as needed in applications ranging from targeted therapy to collective transport and emergency evacuation. In the present work, we consider finite-horizon minimum-energy collective steering for inertial swarms that are subject to stochastic disturbances. We focus on the mean-field representations of these multi-agent systems driven by Cucker--Smale alignment or Morse attraction--repulsion interactions. Our objective is to steer the swarm between prescribed endpoint distributions using a state-feedback control, where the endpoint specifications can be full phase-space distributions (positions and velocities) or position marginals alone. Our formalism is rooted in the theory of Schrödinger bridges, which has inspired contemporary developments spanning statistical inference, biological modeling, stochastic control, and generative learning. Within the bridges framework, the uncontrolled interacting stochastic dynamics are viewed as a prior model, and the optimal control as the minimum-energy corrective drift needed to realize the prescribed distributions. We derive nonlinear, coupled necessary optimality systems with a time-symmetric structure reminiscent of classical Schrödinger bridges, and propose nested fixed-point schemes to numerically solve them. Numerical examples show that the obtained optimal control (corrective drift) can dynamically exploit or counteract the interaction forces, depending on whether the latter are favorable or adversarial to the steering task.

I. INTRODUCTION

The paper formulates finite-horizon minimum-energy steering for noisy, interacting inertial swarms, extending Schrödinger bridges to mean-field kinetics with complete or partial endpoint information. It derives optimality systems and numerical schemes whose controls exploit favorable interactions or counteract adversarial ones.

  • Motivation: Interactions can generate collective behaviors, but may not achieve a desired aggregate configuration within a prescribed time horizon.This motivates controlled steering for applications requiring timely pattern formation or reconfiguration.
  • Problem formulation: The study seeks a broadcast control that guides noisy interacting particles between prescribed initial and target distributions by a chosen terminal time.The control regulates collective dynamics rather than individual trajectories.
  • Schrödinger-bridge perspective: Schrödinger bridges select the smallest perturbation of prior dynamics compatible with endpoint marginals, equivalently interpreting the added drift as minimum expected-energy control.The uncontrolled dynamics serve as the prior evolution.
  • Problem formulation: The models are Cucker–Smale alignment and Morse-potential attraction–repulsion kinetics treated in the mean-field limit through a nonlinear Vlasov–Fokker–Planck PDE.Endpoint specifications may cover full position–velocity phase space or positions alone.
  • Contributions: The work extends Schrödinger bridges to interacting kinetic systems with nonlocal mean-field interactions and complete or partial endpoint observations.It derives nonlinear coupled necessary optimality systems and nested fixed-point schemes for their computation.
  • Schrödinger-bridge perspective: The resulting bridge law retains the Markov property because the prior is an Itô diffusion and the bridge has a multiplicative structure.This connects the classical bridge construction to the paper’s stochastic-control formulation.

B. Stochastic Control Formulation

The stochastic-control formulation recasts the Schrödinger bridge as minimizing the expected energy of an added drift while preserving endpoint constraints. The derivation uses alternative laws, Girsanov’s theorem, and a state-feedback representation.

  • Change of measure: The candidate law is evaluated relative to the prior over the full time-dependent trajectory, with the initial relative-entropy term vanishing when the initial densities coincide.The notation u· denotes the entire trajectory (u_t)_{t∈[0,T]}.
  • Change of measure: Girsanov’s theorem relates the candidate alternative law to the prior diffusion through the added drift and its stochastic likelihood-ratio expression.Taking expectations under the alternative law yields the control-energy term.
  • Entropy and control energy: The bridge minimizes relative entropy with respect to the prior, so its added drift minimizes the corresponding expected control energy.This provides the stochastic optimal-control reformulation of the Schrödinger bridge problem.
  • Feedback control: The optimal control has a state-feedback form determined by the solution of the Schrödinger system.The phase-space density dynamics differ from the prior dynamics by an additional drift involving σ^2∇_x log ϕ_t.

C. Formulation for Kinematic Interacting Particle Systems

The section extends Schrödinger-bridge control from kinematic to interacting inertial systems, formulated in phase space with velocity noise and mean-field coupling. It establishes the kinetic setting and connects it to Cucker–Smale applications.

  • Kinematic interacting systems: Prior work developed Schrödinger bridges for kinematic interacting particles, but interacting kinetic bridge formulations remain much less explored.The kinetic setting introduces inertial positions and velocities governed by coupled stochastic dynamics.
  • Mean-field formulation: In the mean-field limit, empirical particle measures converge to deterministic distributions governed by nonlinear Vlasov–Fokker–Planck or McKean–Vlasov equations.The interaction force depends on the density across the state space and is therefore nonlocal.
  • Mean-field bridge formulation: Unlike classical bridges, the interacting prior depends on the evolving density generated by the path law, making the formulation generally nonconvex.The prior cannot be specified independently of the candidate evolution’s marginals.
  • Kinetic mean-field model: Each inertial particle evolves through dX_t = V_t dt and a velocity equation containing interaction drift and stochastic forcing.The state is the position–velocity tuple, with disturbances acting directly only on velocity.
  • Kinetic mean-field model: The kinetic system can be viewed as a first-order McKean–Vlasov model on the extended state space R2d with degenerate noise.The drift combines velocity transport with the interaction force, while noise enters the velocity component.
  • Cucker–Smale kinetics: The framework covers Cucker–Smale alignment, whose distance-dependent communication weights produce velocity alignment and emergent flocking or schooling.The corresponding mean-field prior is a kinetic Vlasov–Fokker–Planck evolution with position and velocity density µ_t(x,v).
  • Cucker–Smale kinetics: The controlled model adds a state-feedback drift u_t(x,v) to steer prescribed phase-space distributions, while also allowing position-marginal endpoint constraints.The controlled density satisfies a corresponding Vlasov–Fokker–Planck equation.

A. Phase-space Endpoint Constraints

With full phase-space endpoint densities prescribed, the variational derivation yields a nonlinear, coupled Schrödinger system whose controller retains a gradient-feedback form, with gradients taken in velocity. Its forward-backward boundary conditions are coupled through the endpoint constraints, complicating numerical solution.

  • The prescribed endpoint densities fix both position and velocity distributions at the initial and final times.
  • Vanishing first variations with respect to the control and density produce a necessary controller condition and an HJB equation.
  • The Cole–Hopf transformation introduces a positive function φt satisfying a reaction-advection-diffusion equation.
  • The optimal controller preserves the classical Schrödinger bridge’s gradient-feedback structure, but its gradient is taken with respect to velocity rather than position.
  • The resulting optimality system is nonlinearly and nonlocally coupled across time, so its variables cannot be solved independently.The backward and forward integro-PDEs use terminal and initial boundary conditions, respectively, which are coupled by the endpoint constraints and make numerical solution more challenging.
  • A nested iterative computation with delayed evaluations is proposed to solve the coupled system, aided by the adjoint relation between the two evolution equations.

B. Spatial Endpoint Marginal Constraints

The partial-information formulation prescribes only spatial endpoint marginals, leaving endpoint velocity distributions free and extending kinetic Schrödinger bridges to nonlocal interactions. This changes the optimization through an initial relative-entropy term while retaining the controller and evolution structure.

  • The spatial-marginal formulation addresses endpoint steering when only partial information about the system is available.
  • Prescribing only the initial spatial marginal leaves freedom in choosing the initial velocity distribution subject to the marginal constraint.
  • The path-space formulation uses Girsanov’s theorem to relate controlled dynamics to the prior dynamics.
  • Unlike the full phase-space setting, the relative entropy H(µ0, ν0) does not vanish and contributes to the optimization criterion.
  • Because endpoint phase-space densities are not fully fixed, their perturbations may be nonzero in the variational derivation.
  • The optimality system retains the same controller, HJB equation, and transformed evolution as the full phase-space formulation, but changes its endpoint constraints.The terminal multiplier is independent of velocity, producing a distinct terminal control structure.

IV. BRIDGES OVER MORSE-POTENTIAL DRIVEN KINETICS

The Morse-potential extension applies the bridge-based inference-control framework to inertial swarms with attractive and repulsive interactions. Its mean-field force depends on the spatial marginal, while the resulting optimality system modifies the interaction contribution but otherwise follows the established structure.

  • The model considers stochastic inertial particles with attractive and repulsive pairwise interactions governed by a Morse potential.
  • The Morse interaction combines short-range repulsion with long-range attraction, mitigating collisions while helping maintain collective cohesion.
  • In the mean-field limit, the Morse force depends exclusively on the spatial marginal and the position coordinate.
  • Given full initial and final phase-space densities, the Morse-driven system is formulated as an inference-control problem with fixed endpoint distributions.
  • The optimal controller retains its general form, while the Cucker–Smale interaction-dependent term is replaced by a Morse-specific contribution.
  • Optimizing over the density yields an HJB equation, and the resulting forward-backward optimality system is stated with coupled endpoint conditions.
  • The same framework is extended to partial-information steering with prescribed initial and final spatial marginals.

V. EXAMPLES

The numerical examples show how controlled Cucker–Smale dynamics steer a disordered swarm toward spatial splitting and prescribed marginals. The controller first opposes alignment to accelerate separation, then exploits the interaction to brake the swarm, achieving the lowest accumulated cost among the compared controls.

  • A. Steering the Cucker–Smale Kinetics Towards a Phase-Space Distribution: The example starts with two disordered subpopulations moving in opposite directions and requires spatial splitting with aligned final velocities.More than 99.9% of the velocity mass lies in [−2.8, 2.3], with concentrations near −1.5 and 1.
  • A. Steering the Cucker–Smale Kinetics Towards a Phase-Space Distribution: Without control, Cucker–Smale alignment gradually synchronizes the two subpopulations and eventually makes them coalesce in phase space.The uncontrolled streamlines represent the instantaneous phase-space drift and show acceleration below the mean velocity and deceleration above it.
  • A. Steering the Cucker–Smale Kinetics Towards a Phase-Space Distribution: The computed controller accelerates the subgroups against the interaction to facilitate spatial separation, then works with the interaction to brake the nearly split swarm.The control effect is assessed through the weighted inner product of the controller and the Cucker–Smale forcing.
  • A. Steering the Cucker–Smale Kinetics Towards a Phase-Space Distribution: The computed spatial and velocity marginals match the prescribed endpoint marginals.FIG. 2a and FIG. 2b show the evolution of the two marginal distributions.
  • A. Steering the Cucker–Smale Kinetics Towards a Phase-Space Distribution: Accumulated costs are 1.34 for the computed interacting controller, 2.69 for the non-interacting controller, and 2.07 for the baseline controller.The interacting controller is cheapest because it uses the interaction force when advantageous.

B. Inference with the Cucker–Smale Kinetics and Partial Information

With only spatial endpoint information, the Cucker–Smale and Morse examples infer compatible phase-space evolutions and produce controllers that match prescribed marginals. The controller benefits from favorable alignment but must spend more energy when counteracting repulsive interactions, while the derived solutions are stationary points rather than guaranteed global minima.

  • B. Inference with the Cucker–Smale Kinetics and Partial Information: The partial-information Cucker–Smale problem infers a drift steering prescribed spatial marginals without constraining the final velocity distribution.The initial velocity distribution is selected through the prior, while the final velocity distribution is consequently less concentrated than in the full phase-space example.
  • B. Inference with the Cucker–Smale Kinetics and Partial Information: Accumulated costs are 0.24 for the computed interacting controller, 0.63 for the non-interacting controller, and 2.71 for the baseline controller.The interacting controller again has the lowest cost among the compared controllers.
  • C. Steering the Morse-potential-driven Dynamics Towards a Target Phase-Space Distribution: The uncontrolled Morse interaction disperses the initially concentrated population through its repulsive component.The model combines short-range repulsion with long-range attraction.
  • C. Steering the Morse-potential-driven Dynamics Towards a Target Phase-Space Distribution: The Morse target is more spatially concentrated and velocity-aligned than the initial distribution, representing a cohesive flock.The controller counteracts the interaction for most of the evolution to draw the population closer together, while the computed marginals match the specified endpoints.
  • C. Steering the Morse-potential-driven Dynamics Towards a Target Phase-Space Distribution: Accumulated costs are 2.96 for the computed interacting controller, 1.94 for the non-interacting controller, and 3.71 for the baseline controller.The interacting controller costs more than the non-interacting controller because it must counteract the repulsive interaction.
  • VI. CONCLUSIONS: The nested fixed-point approach solves the nonlinear systems, while the nonconvex optimization yields stationary points rather than guaranteed global minima.The paper identifies second-order conditions as a future direction.

Appendix A: Iterative Approaches for Solving the Necessary Optimality Systems

The numerical method uses nested fixed-point iterations: an inner solve holds the mean-field density fixed, while an outer update refreshes it. Delayed nonlinear evaluations, damping, and adjoint linear solves support computation and convergence.

  • The scheme first solves a non-interacting second-order Schrödinger bridge using a Klein–Kramers Sinkhorn iteration.
  • The inner iteration solves the forward-backward equations with the density and nonlocal drift held fixed, using delayed evaluations for nonlinear terms.
  • The outer iteration updates the density, thereby modifying the dependent mean-field drift.
  • The Hilbert metric provides convergence criteria for the positive functions φt, ˆφt, and µt.
  • When drift and reaction rate are fixed, the resulting forward and backward PDEs are linear and mutually adjoint, simplifying numerical integration.
  • Relaxed density updates with damping parameter θ ∈ (0, 1] can improve numerical stability when updates are overly aggressive.

2. Cucker–Smale with Spatial Endpoint Marginals

For Cucker–Smale dynamics with spatial endpoint constraints, the iterative construction is initialized from a non-interacting bridge and solved through Sinkhorn-type updates.

  • The spatially constrained bridge can be initialized using a non-interacting Schrödinger bridge with a = 0.
  • The non-interacting initialization satisfies the spatial endpoint-marginal constraints through the pair (ϕ·, ˆϕ·).
  • Numerical solutions for the initialization are obtained via a Sinkhorn iteration.

3. Morse Interaction with Phase-Space or Spatial Endpoint Marginals

The Morse-interaction treatment adapts the iterative and adjoint numerical machinery to interaction-dependent drift, while exploiting the velocity-independent structure of the Morse force.

  • Because the Morse interaction term is independent of velocity, its forward advection simplifies to an inner product involving ∇v ˆφt.
  • The same adjoint-based integration strategy applies to the forward and backward equations in the Morse-interaction schemes.
  • The discrete formulation represents φt and ˆφt as vectors and the propagators as matrices, with one propagator obtained from the transpose of the other.
  • Strang splitting approximates the discrete forward propagator by separating advection, diffusion, and reaction operators.

2. Strang Splitting

The numerical implementation uses Strang splitting for the Vlasov–Fokker–Planck propagator, combining operator decomposition with semi-Lagrangian advection.

  • Strang splitting provides a second-order approximation for the discrete forward propagator of the Vlasov–Fokker–Planck equation.
  • The advection step computes characteristic trajectories with a semi-Lagrangian method and can use explicit Runge–Kutta or Verlet schemes.
  • For Morse-potential-driven kinetics, the forward propagator is approximated directly and the backward propagator by transposition.
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