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Equilibrium Propagation for Learning in Lagrangian Dynamical Systems

Serge Massar

arXiv:2505.07363v3nlin.CDcs.LGphysics.data-an

TL;DR

Training dynamical systems with Lagrangian mechanics motivates an alternative to backpropagation through time that can update parameters together. The paper adapts Equilibrium Propagation by nudging trajectories and estimating updates from perturbed solutions; two trajectories yield all parameter updates, under suitable boundary conditions.

  • Problem

    Backpropagation through time is a straightforward training method for time-dependent networks, motivating more efficient updates of all parameters and methods suited to mechanical dynamical systems.

    Source: §I · p. 1

  • Method

    The method nudges Lagrangian trajectories through a cost term and estimates parameter updates from the response of perturbed trajectories, without explicitly propagating backward in time.

    Source: §I · pp. 1–2, §III · p. 5

  • Results

    Two oppositely nudged trajectories suffice to estimate all parameter updates, while periodic boundary conditions yield the semiclassical limit of Quantum Equilibrium Propagation.

    Source: §I · p. 2, §III · p. 5

  • Takeaways & Limitations

    Periodic boundary conditions connect the method to semiclassical Quantum Equilibrium Propagation and can accommodate weak dissipation effects.

    Source: §V · p. 8

    The method applies only to selected trajectory boundary conditions; ordinary specified initial position and velocity do not work.

    Source: §I · p. 1, A. Causal boundary conditions do not… · p. 6

Abstract

from arXiv · show

We propose a method for training dynamical systems governed by Lagrangian mechanics using Equilibrium Propagation. Our approach extends Equilibrium Propagation - initially developed for energy-based models - to dynamical trajectories by leveraging the principle of action extremization. Training is achieved by gently nudging trajectories toward desired targets and measuring how the variables conjugate to the parameters to be trained respond. This method is particularly suited to systems with periodic boundary conditions or fixed initial and final states, enabling efficient parameter updates without requiring explicit backpropagation through time. In the case of periodic boundary conditions, this approach yields the semiclassical limit of Quantum Equilibrium Propagation. Applications to systems with dissipation are also discussed.

I. INTRODUCTION

The paper extends Equilibrium Propagation to Lagrangian dynamical systems, where equations of motion extremize classical action. It uses trajectory nudging to update parameters, but requires boundary conditions that let the nudge affect the trainable region.

I. INTRODUCTION

  • Lagrangian dynamics can be trained with Equilibrium Propagation without explicitly propagating the system backward in time, but only for selected trajectory boundary conditions [9].The method applies to periodic trajectories or trajectories with fixed initial and final conditions; it also allows some forms of dissipation.Source: §I · p. 1
  • Equilibrium Propagation adapts to dynamics by nudging trajectories toward targets and using the modified trajectory to compute parameter updates.The method generalizes the equilibrium condition to equations of motion that extremize classical action.Source: §I · pp. 1–2
  • Ordinary initial-position-and-velocity conditions do not work when the nudge leaves the region of the trajectory where trainable forces act unchanged.Fixed initial and final positions, fixed initial and final velocities, and periodic conditions instead allow a force applied along the trajectory to modify it everywhere.Source: §I · p. 2
  • The ordering of input, trainable, and nudging forces is not essential, since they can act simultaneously.Source: §I · p. 2
  • The direction of the cost-reducing trajectory displacement indicates whether to increase or decrease the trainable force.In the illustrated fixed-position and fixed-velocity cases the force decreases; in the periodic case it increases.Source: §I · p. 2
  • Periodic boundary conditions are especially interesting because they yield the semiclassical limit of Quantum Equilibrium Propagation and permit incorporating dissipation.The paper also describes periodic boundary conditions as especially natural in applications.Source: §I · p. 2

A. General formulation

The formulation adds a weighted trajectory cost to a parameterized Lagrangian. Solutions extremize the resulting action subject to boundary conditions, and training seeks parameters that reduce average cost.

A. General formulation

  • The Lagrangian is augmented by β times the cost, so β nudges the trajectory toward or away from the desired behavior.The cost is defined as an integral over time of a density measuring deviation from the desired trajectory.Source: A. General formulation · p. 2
  • The system trajectory is a solution of the Euler–Lagrange equations for the chosen Lagrangian and boundary conditions.The solution is denoted with a subscript *; quantities evaluated on it, including the Lagrangian, carry that subscript.Source: A. General formulation · pp. 2–3
  • The action on a solution is the action functional evaluated along that trajectory, while the cost on a solution is its trajectory-integrated cost.Source: A. General formulation · p. 3
  • Training adjusts parameters α to minimize average cost over inputs when β is zero.Source: A. General formulation · p. 3

B. Illustrative example

The illustrative model combines an unperturbed dynamical system with input-dependent and trainable forces. A trajectory cost attracts selected outputs toward targets that may be static or time-varying.

B. Illustrative example

  • The example's trainable forces arise from parameter-dependent potential terms, while input forces come from an input-dependent potential.The trainable potential uses time-dependent weights and a sigmoid of the state; the input potential can couple the system to an input trajectory.Source: B. Illustrative example · p. 4
  • The example's input potential can represent image pixels as forces and can clamp selected state variables to input values.Clamping is obtained by taking the input spring constant arbitrarily large.Source: B. Illustrative example · p. 4
  • The target cost measures distance between the system trajectory and a desired trajectory, with the corresponding nudging force given by the negative cost gradient.The cost acts on readout variables and may be weighted over time.Source: B. Illustrative example · p. 4
  • Targets can be static for classification or time-dependent, such as a sinusoid whose frequency encodes the output.For MNIST, the example uses 10 outputs with one-hot encoding.Source: B. Illustrative example · p. 4
  • With the kinetic term neglected and all time-dependent functions set to one, the example reduces identically to the problem of.The nontrivial time dependence is the stated novelty of this example.Source: B. Illustrative example · p. 4

III. EQUILIBRIUM PROPAGATION

The update rule estimates parameter gradients by comparing trajectories under positive and negative cost nudges. This requires a vanishing boundary term for the stated finite-difference estimate.

III. EQUILIBRIUM PROPAGATION

  • Iterating the parameter updates decreases the expected cost.Source: §III · p. 5
  • The finite-difference estimate of the cost derivative requires the boundary term to vanish.The procedure computes trajectories for small positive and negative β and estimates the derivative from their difference.Source: §III · p. 5
  • Two trajectories nudged by positive and negative cost values provide estimates of all parameter gradients and hence all parameter updates.The result is the dynamical-systems analog of the Equilibrium Propagation update rule.Source: §III · p. 5
  • For a parameter entering the Lagrangian through a potential linear in that parameter, the update direction follows whether the cost-reducing displacement aligns with its force.Increase the parameter when the displacement is along the trainable force, and decrease it when the displacement is opposite.Source: §III · pp. 5–6
  • The update integrand can be interpreted as minus the work induced by cost nudging, per unit nudging and per unit parameter.Source: §III · p. 6

A. Causal boundary conditions do not work

The gradient relation is valid only when the boundary terms vanish, and specifying the initial position and velocity does not satisfy the needed conditions. In that setting, boundary terms can determine the parameter gradient entirely.

A. Causal boundary conditions do not work

  • Equation (19) is valid only when its boundary terms vanish, which requires appropriate boundary conditions.The section treats boundary conditions as necessary for using the gradient relation.Source: A. Causal boundary conditions do not… · p. 6
  • Specifying the initial position and velocity does not provide boundary conditions that make the method work.The example in Figure 1(a) illustrates this failure.Source: A. Causal boundary conditions do not… · p. 6
  • The parameter gradient can be entirely due to boundary terms when the first term in Equation (19) vanishes and ∂L*/∂α_j is independent of β.This occurs, for example, when the trainable forces and nudging are applied in the order shown in Figure 1(a).Source: A. Causal boundary conditions do not… · p. 6

B. Fixed initial and final position

Fixing both the initial and final positions is the simplest boundary condition identified here for eliminating the boundary terms in Equation (17). This case is shown in Figure 1(b) and was introduced in [9].

B. Fixed initial and final position

C. Fixed initial and final velocity

Fixed endpoint velocities eliminate the first boundary term, but a further condition on the Lagrangian is needed to eliminate the second. Periodic boundary conditions are also considered, with the trajectory and Lagrangian extended periodically.

C. Fixed initial and final velocity

  • Fixing initial and final velocities removes the first boundary term, but the second requires the Lagrangian's dependence on β and α_j to vanish at both endpoints.The endpoint conditions include fixed velocities and vanishing mixed derivatives with respect to velocity and β.Source: C. Fixed initial and final velocity · p. 6
  • The additional endpoint conditions hold when trainable Lagrangian terms and the cost are velocity-independent, or when their α-dependent terms vanish at both endpoints.This case is depicted in Figure 1(c).Source: C. Fixed initial and final velocity · p. 6

1. General condition

  • Periodic boundary conditions are described by matching the trajectory and Lagrangian after a period T: x*(t + T) = x*(t) and L(x, ẋ, t + T) = L(x, ẋ, t).The paper presents periodic conditions as particularly interesting.Source: §1 · p. 7
  • The periodic formulation extends the trajectory and Lagrangian to periodic functions with t_f = t_i + T.The periodic case is illustrated in Figure 1(d).Source: §1 · p. 7

2. Time independent case and Hamiltonian formalism

For time-independent Lagrangians, periodic trajectories can be paired with different constraints. The fixed reduced-action choice is identified as the semiclassical version of Quantum Equilibrium Propagation.

2. Time independent case and Hamiltonian formalism

  • A constant solution with ẋ* = 0 places the system at a local energy minimum and recovers the original problem of.In this case, time dependence disappears.Source: §2 · p. 7
  • For periodic trajectories at fixed energy E, the period T is free to adjust to satisfy the energy constraint.Source: §2 · p. 7
  • Fixing the reduced action S0 gives the semiclassical version of Quantum Equilibrium Propagation [13] [14].The period T adjusts to satisfy the fixed-action condition; the stated quantization condition includes an integer n and the Maslov index α.Source: §2 · p. 7

E. Damped equations of motion

The damped formalism multiplies the original Lagrangian by an exponential factor, adding damping to the equations of motion. For periodic trajectories, damping leaves boundary terms that become negligible as damping decreases.

E. Damped equations of motion

  • Damping breaks time-reversal symmetry in the equations of motion.Source: E. Damped equations of motion · p. 8
  • Multiplying the original Lagrangian by e^Γt produces equations of motion with an added damping term.The formalism considers this specific way of introducing damping.Source: E. Damped equations of motion · p. 8
  • With fixed initial and final positions or velocities, Equilibrium Propagation still applies, with time-dependent functions γ(t) appearing in Eq. (20).Source: E. Damped equations of motion · p. 8
  • For periodic trajectories, damping makes the new Lagrangian nonperiodic, so boundary terms do not vanish; when ΓT ≪ 1, neglecting them incurs less error as damping decreases.The boundary term B(t) is computed for the original Lagrangian.Source: E. Damped equations of motion · p. 8

V. CONCLUSION

The paper extends Equilibrium Propagation to Lagrangian dynamical systems, estimating all parameter updates from two perturbed trajectories. Its applicability depends on boundary conditions, with periodic trajectories connecting to quantum Equilibrium Propagation’s semiclassical limit and accommodating weak dissipation.

V. CONCLUSION

  • Two perturbed trajectories suffice to estimate all parameter updates in the Lagrangian Equilibrium Propagation framework.The cost is expressed as a functional that can be included in the Lagrangian.Source: §V · p. 8
  • Successful application requires suitable boundary conditions, such as periodic trajectories or trajectories with fixed initial and final states.Source: §V · p. 8
  • Periodic boundary conditions connect the approach to the semiclassical limit of quantum Equilibrium Propagation and can accommodate weak dissipation.Source: §V · p. 8
  • The authors conclude that Equilibrium Propagation extends beyond static energy-based models to dynamical systems with time-dependent or dissipative effects.They identify physical, analog, and neuromorphic computing as potential applications, and list stochastic dynamics and Hamiltonian-based approaches among future directions.Source: §V · p. 8

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