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LeHami: Symplectic Latent Dynamics for World Models

Romina Yalovetzky, Randall Balestriero, Yann LeCun

arXiv:2610.15267v1cs.LGcs.AIcs.RO

TL;DR

Unconstrained JEPA rollouts can contract latent distributions over time. LeHami replaces the standard predictor with an action-conditioned Hamiltonian flow over canonical (q, p), integrated by an exact symplectic leapfrog step. On the conservative pendulum, it leads LeWM in velocity prediction by 2.8× over the first ten steps and 1.3× at the horizon end.

  • Problem

    Unconstrained autoregressive JEPA predictors can compound per-step contraction, shrinking latent distributions over long rollouts despite approaches that prevent encoder collapse.

    Source: §1 · p. 1

  • Method

    LeHami evolves a canonical latent pair with an action-conditioned Hamiltonian flow and exact symplectic leapfrog integration, deriving momentum from the temporal derivative of the position-like representation.

    Source: §1 · p. 2, §2 · p. 3, §2.1 · p. 5

  • Results

    2.8× over the first ten steps and 1.3× at the horizon end: LeHami’s velocity prediction leads LeWM on the conservative pendulum.

    Source: §3.2 · p. 7

  • Takeaways & Limitations

    Symplecticity provides an inference-time volume-preservation guarantee complementary to encoder regularization, but does not preserve distributional shape or bound rollout-error growth.

    Source: §2 · p. 3

    Exact volume preservation is a scope boundary for strongly dissipative systems, whose phase-space volume contracts; extending LeHami to such regimes remains future work.

    Source: §4 · p. 10

Abstract

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Joint-embedding predictive architectures (JEPAs) learn world models directly in latent space, but unconstrained autoregressive predictors may contract the latent distribution over long rollouts. We introduce Symplectic Hamiltonian JEPA (LeHami), which replaces the standard predictor with an action-conditioned Hamiltonian flow over a structured latent state (q, p) integrated using an exactly symplectic leapfrog step. This construction preserves phase-space volume at every rollout step and rules out uniform contraction, asymptotically stable fixed points, and merging of distinct latent states, independently of the learned parameters or action sequence. We obtain the momentum-like component from temporal derivatives of the learned position-like representation, providing an explicit second-order state for prediction and control. On conservative and actuated visual pendulum environments, LeHami substantially improves long-horizon velocity prediction over an unconstrained JEPA world model and empirically maintains constant latent volume where the baseline contracts. The learned flow also supports accurate integration at unseen sub-frame step sizes and numerical time reversal. These results show how geometric structure can provide inference-time guarantees that are complementary to representation regularization in latent world models.

1 INTRODUCTION

Unconstrained JEPA rollouts can progressively shrink latent distributions, while existing approaches address encoder collapse but leave long-horizon dynamics challenging. LeHami applies an action-conditioned Hamiltonian flow to a canonical latent state, providing structural rollout guarantees and representing position and momentum together.

1 INTRODUCTION

  • Unconstrained autoregressive predictors can compound slight per-step contraction, progressively shrinking latent distributions over long rollouts despite approaches that prevent encoder collapse.Those approaches use training-time regularization or frozen backbones.Source: §1 · p. 1
  • LeHami combines an action-conditioned Hamiltonian flow over canonical latents with an exact symplectic step, trained end-to-end without decoders or supervised coordinates.The latent state is the pair (q, p).Source: §1 · pp. 2–3
  • The Hamiltonian flow preserves phase-space volume under actuation and supports arbitrary integration step sizes and backward-time rollouts.The action enters the Hamiltonian; flipping time enables backward-in-time rollouts.Source: §1 · p. 2
  • LeHami’s canonical phase space represents position and momentum together, which the introduction identifies as necessary to uniquely determine future physical evolution.Standard JEPA architectures otherwise rely on design choices such as longer encoder windows or context-conditioned predictors to capture velocity, without guarantees.Source: §1 · p. 2
  • Across conservative and actuated pendulum environments, LeHami recovers full system dynamics and achieves numerical time-reversal round-trip error below 10^-4%.The authors describe full-state prediction as a basis for planning toward goals specifying position and velocity.Source: §1 · p. 3

2 LEHAMI: SYMPLECTIC HAMILTONIAN JEPA

LeHami uses an action-conditioned Hamiltonian flow to evolve a canonical latent pair, with exact symplecticity preserving phase-space volume at every rollout step. Its architecture derives momentum from the encoder trajectory, trains multi-step predictions with a regularizer, and differs from related latent Hamiltonian approaches in its canonical state and symplectic rollout.

2 LEHAMI: SYMPLECTIC HAMILTONIAN JEPA

  • The leapfrog step is an exactly symplectic diffeomorphism for every step size and parameter, preserving phase-space volume and ruling out uniform contraction, asymptotically stable fixed points, and merging distinct states.The action is held fixed within each step; the step size need not equal the observation interval at inference.Source: §2 · p. 3
  • Volume preservation does not bound rollout-error growth or preserve distributional shape, so symplecticity does not make encoder regularization redundant.Regularization acts on the encoder distribution in expectation over batches, whereas symplecticity holds at every inference step.Source: §2 · p. 3

2.1 THE ARCHITECTURE

  • Under actuation, the action conditions the potential, making the generalized force a q-gradient while preserving exact symplecticity for each action sequence.The action must remain constant within each step.Source: §2.1 · p. 5
  • Training iterates the Hamiltonian flow for K self-fed steps, averaging position- and momentum-embedding prediction losses with a regularizer on ground-truth embeddings.The regularizer may be applied to q and p independently or together, and never to the rollout.Source: Figure 2 · p. 4, §2.1 · p. 5
  • A fixed Savitzky–Golay stencil derives momentum p as the temporal derivative of position-like q, using a least-squares quadratic fit without learned parameters.The architecture uses the encoder trajectory to form the canonical pair (q, p).Source: §2 · p. 3, §2.1 · p. 5
  • The work uses a separable Hamiltonian so leapfrog integration is explicit and avoids iterative solvers; non-separable Hamiltonians can retain exact symplecticity with a suitable integrator.A constant linear rescaling of q and p also leaves the flow exactly symplectic.Source: §2.1 · pp. 4–5

2.2 RELATED WORK

  • Unlike related JEPA approaches that use an external control term or advance only a slice of a wider code, LeHami uses an action-conditioned canonical (q, p) state and an exactly symplectic rollout.The passage contrasts it with H-JEPA’s explicit Euler step, which is not symplectic.Source: §2.2 · p. 5

3 EMPIRICAL VALIDATION OF LEHAMI

LeHami is evaluated on conservative and actuated visual pendulums using frozen probes, long rollouts, and goal-reaching tasks. It improves velocity prediction and preserves latent volume, while supporting time re-stepping and reversal.

3.1 EXPERIMENTAL SETUP

  • Evaluation uses rendered 64×64 RGB pendulums, frozen probes, and goal-reaching planning; the actuated environment varies torque, rod length, and mass.The conservative environment has no actuation and varies rod length; planning optimizes actions under a strict computational budget.Source: §3 · p. 5, §3.1 · p. 6

3.2 EVALUATING COMPLETE PHASE-SPACE PHYSICS CAPTURE

  • On the conservative pendulum, LeHami leads LeWM in velocity prediction by 2.8× over the first ten steps and 1.3× at the horizon’s end.LeHami’s normalized cumulative log-volume stays at 0 (|det J(zi)| = 1), while LeWM’s declines steadily during rollout.Source: §3.2 · p. 7, Figure 4 · p. 7
  • LeHami has lower θ and ω error than LeWM at every matched training budget, matching LeWM’s best error with at most one-eighth of the training FLOPs.The advantage also holds with a frozen pre-trained encoder: DINO-LeHami outperforms DINO-WM.Source: §3.2 · p. 7
  • On the actuated pendulum, LeHami maintains superior velocity prediction until K ≈10; LeWM initially decodes angle more accurately, but that advantage decays by K ≈40.At the anchor, nonlinear probes decode LeHami’s velocity well, while all three LeWM probe tiers fall below the predict-the-mean floor for velocity.Source: §3.2 · p. 7, Table 1 · p. 8

3.3 LEHAMI IS A WORLD MODEL WITH TIME RE-STEPPING AND REVERSAL

  • Halving or quartering the trained ∆t = 0.1 s step traces the same trajectory, doubling it breaks, and forward-then-backward rollouts have round-trip error below 10^-4%.The step-size comparison uses ground truth from integration of the ODE; the model is held fixed during re-stepping.Source: §3.3 · p. 8, Figure 5 · p. 8

3.4 LEHAMI BRINGS COMPUTATIONAL EFFICIENCY AND STRUCTURAL PRIORS IN VELOCITY PLANNING

  • Under the 15 TFLOP-per-episode budget, untied-window LeWM achieves a higher success rate, while LeHami’s lower compute cost allows more trajectories and more frequent replanning.LeHami uses R=6 steps with five solves on velocity and joint clauses; the passage reports no success-rate values.Source: §3.4 · p. 9, Table 2 · p. 9

4 CONCLUSION

LeHami preserves latent phase-space volume and supports continuous-time integration with time reversal, while improving long-horizon predictions in the reported pendulum settings. The conclusion also identifies scope boundaries and points to future work.

4 CONCLUSION

  • LeHami preserves latent volume while LeWM contracts it on the conservative pendulum; LeHami predicts velocity better throughout and angle better at long horizons.Under actuation, LeWM reaches the goal angle more often, but captures signed velocity only with an explicitly untied temporal window; LeHami captures angle and signed velocity by construction.Source: §4 · p. 9
  • Strongly dissipative systems, static-pose tasks, and the current zero-overhead SG stencil mark distinct scope boundaries or future-work directions.The authors identify port-Hamiltonian extensions for heavy friction, spatial-focused predictors for static pose, and learned momentum-like embeddings as future work.Source: §4 · p. 10

A APPENDIX ORGANIZATION

  • The appendices collect Hamiltonian background and formal guarantees, then document design choices, evaluation protocols, and supplementary pendulum results.Supplementary results include prediction, planning, latent-volume, encoder-independence, and compute analyses for the conservative and actuated settings.Source: A APPENDIX ORGANIZATION · pp. 12–13

B BACKGROUND IN HAMILTONIAN MECHANICS

  • Hamiltonian flow preserves the symplectic form and therefore phase-space volume, while transporting density without distortion.A cloud of states may stretch, rotate, and shear, but its volume remains invariant.Source: B BACKGROUND IN HAMILTONIAN MECHANICS · p. 13

C.1 SYMPLECTICITY OF THE LEAPFROG STEP: FORMAL STATEMENT AND PROOF

  • The formal proof establishes the leapfrog map’s symplecticity through its half-step maps and their composition.The formal statement assumes a separable Hamiltonian and holds the control fixed within each step.Source: §C.1 · p. 13

A C1 map Ψ : R2d →R2d is symplectic if DΨ(z)⊤J DΨ(z) = J for every z, where DΨ(z) ∈ R2d×2d denotes the Jacobian of Ψ at z, i.e., [DΨ(z)]ij = ∂Ψi/∂zj.

The leapfrog map is an exactly symplectic diffeomorphism, so its iterates preserve volume, exclude asymptotically stable fixed points, and keep distinct latent states distinct. The proof builds this result from symplectic half-steps and their composition.

Formal guarantee

Proof strategy

  • The proof establishes exact symplecticity by composing symplectic half-steps, each of which is a diffeomorphism with unit determinant.The half-steps are shown to be symplectic shears, and symplecticity is preserved under composition.Source: A C1 map Ψ : R2d →R2d is symplectic if… · p. 14

C.2 SCOPE AND LIMITATIONS OF THE GUARANTEE

Symplecticity rules out specific degenerate rollout failures, but it does not guarantee accurate predictions or replace distributional regularization. The paper uses Hamiltonian structure as a geometric prior, not as a model of the observed system’s physical laws.

C.2 SCOPE AND LIMITATIONS OF THE GUARANTEE

  • Volume preservation prevents uniform squeezing but does not bound rollout-error growth or guarantee accuracy.Latent clouds may still be stretched or sheared arbitrarily.Source: §C.2 · p. 15
  • Symplecticity preserves volume, not distributional shape, so SIGReg and the geometric constraint serve complementary roles.SIGReg shapes encoder distributions during training; symplecticity carries volume at each inference step.Source: §C.2 · p. 15
  • LeHami’s Hamiltonian is not fit to recover the observed system’s physical Hamiltonian; physical equations serve only as evaluation targets for frozen probes.This distinguishes its geometric latent-space prior from Hamiltonian neural networks and SRNN, which seek a surrogate physical Hamiltonian.Source: §C.2 · p. 15
  • The canonical latent pair is structural: q and p are not assumed to encode observation-space position and momentum.The prior restricts JEPA predictors to symplectomorphisms reachable through a leapfrog step of a learned separable generator.Source: §C.2 · p. 15

C.3 WHY HAMILTONIAN AND NOT LAGRANGIAN

  • The Hamiltonian formulation gives a phase-space volume guarantee and an explicit leapfrog step, unlike the Lagrangian formulation’s second-order equations in q.The separable Hamiltonian keeps the leapfrog map exactly symplectic at any step size.Source: §C.3 · p. 16

C.4 DESIGN CHOICES IN THE FLOW AND THE ENCODER

LeHami uses a separable, action-conditioned Hamiltonian and derives its momentum-like state from temporal differences of learned positions. The section explains the stencil, scaling and regularization choices, including the boundary imposed by configuration-dependent mass.

Hamiltonian structure

  • The separable Hamiltonian assigns momentum to the kinetic term and position plus action to the potential, enabling leapfrog integration with action-dependent forces.More general Hamiltonians are possible, but the work focuses on the separable form.Source: §C.4 · p. 16

Momentum from the encoder trajectory

  • A fixed Savitzky–Golay derivative gives p at the same instant as q without learned parameters, while fitting several frames attenuates noise amplification at the cost of truncation bias.The least-squares fit uses fixed coefficients determined by the window and polynomial order.Source: §C.4 · pp. 16–17

Stencil windows

  • The rollout anchor uses a causal four-frame stencil, while centered five-frame windows provide training targets; centered weights reverse p's sign exactly under time reversal.Centered targets are not fed into the rollout, so their use does not leak future frames into the anchor.Source: §C.4 · p. 17

Mass and scope

  • Defining p as q's time derivative fixes the learned-coordinate mass to identity, but a configuration-dependent mass matrix cannot be removed by a global linear coordinate change and makes separable K(p) mis-specified.The text identifies jointly learning p, rather than reading it from a fixed stencil, as the route to lifting this limitation.Source: §C.4 · p. 17

Momentum scaling

  • The learned diagonal scale S accommodates the unconstrained relative scale of q and p while preserving exact symplecticity because it enters only the position update.The rescaling of p can be absorbed into K's input weights, leaving S close to identity in trained models.Source: §C.4 · pp. 17–18

Regularization

  • Step-wise SIGReg is applied across time because distributional regularization does not replace the volume preservation supplied by the symplectic structure.The section describes both joint q,p and separate q- and p-block regularization variants.Source: §C.4 · p. 18

D.1 METHODOLOGY

The methodology compares LeHami with LeWorldModel under shared data and training conditions, differing in predictor and momentum stencil. Evaluation uses held-out episodes and eight self-fed rollout steps.

Controlled comparison

  • The models share the encoder, projector, action encoder, optimizer, data pipeline, objective, epochs, and learning-rate schedule; only the predictor and LeHami's SG stencil differ.Both models receive identical data and clip construction, controlling for training-scheme differences in rollout behavior.Source: §D.1 · p. 18

Evaluation protocol

  • Performance is assessed on test episodes after K = 8 self-fed prediction steps from an anchor, with checkpoints selected on validation data.The datasets are split into train, validation, and test at the episode level.Source: §D.1 · p. 19

Model comparison

  • Both models use 64 latent dimensions, split into 32-dimensional q and p blocks for LeHami, while LeWM uses all 64 dimensions as a single latent.LeWM uses a three-latent causal transformer; LeHami evolves one state at a time with its Hamiltonian flow.Source: §D.1 · p. 19

D.2 IMPLEMENTATION OF LEHAMI

The implementation combines a separable Hamiltonian flow with a temporal ConvNet option and fixed Savitzky–Golay momentum estimates. The conservative pendulum data cover multiple lengths and both oscillating and rotating trajectories.

D.2 IMPLEMENTATION OF LEHAMI

  • LeHami uses two three-layer tanh MLPs for separable K(p) and V(q,u), advanced by one exactly symplectic kick–drift–kick step per frame.The temporal ConvNet factors spatiotemporal processing into a per-frame stem, one temporal mix, and the unchanged spatial trunk; the main model uses W = 1.Source: §D.2 · p. 19, Figure 6 · p. 20
  • The step-size input could support mixed recording rates by expressing each episode's spacing relative to a reference rate, but the paper reports no mixed-rate comparison.The authors state that all models were trained and evaluated at a single rate.Source: §D.2 · p. 19
  • Fixed Savitzky–Golay stencils derive momentum from encoder positions: a causal window initializes the rollout, while centered windows supply targets and evaluation references.The anchor uses only current and preceding frames to avoid reading frames the model is about to predict; no stencil is applied after the anchor.Source: §D.2 · p. 21
  • The reversal diagnostic is approximate because its anchor momentum comes from a causal stencil, which reverses into a forward stencil rather than the negative of itself.The centered stencil, by contrast, changes p to −p exactly under time reversal.Source: §D.2 · p. 21

D.3.1 CONSERVATIVE PENDULUM.

  • The dataset uses three rod lengths (0.7, 1.0, and 1.4 m), rendered to scale, and spans both oscillating and rotating regimes through independently sampled initial conditions.It contains 300 episodes of 80 frames at Δt = 0.1 s, split 210/45/45 across train/validation/test; the three lengths span a 1.41× natural-frequency range.Source: §D.3.1 · p. 21

D.3.2 ACTION AS A FORCE: ACTUATED PENDULUM.

The actuated pendulum adds torque to the conservative dynamics, so energy is not conserved. Episodes vary in rod length and mass, with recorded torque driving each reported rollout.

Actuated pendulum setup

  • The actuated environment adds applied torque to the conservative pendulum dynamics, and reported rollouts use the recorded torque sequence.The episodes vary across rod lengths of 0.7, 1.0, and 1.4 m and masses of 0.5, 1.0, and 2.0 kg.Source: §D.3.2 · pp. 21–22

Actuated dynamics

  • Under actuation, energy is not conserved and the potential carries a time-varying input; mass affects the torque response, so the latent must carry mass signal to predict that response.The data comprise 630/135/135 train/validation/test episodes at 64 × 64 RGB resolution.Source: §D.3.2 · p. 22

Torque input

  • Torque is held constant for h = 3 frames (0.3 s), making the drive observable from pixels and coherent over a fraction of a pendulum swing.The hold spans 13–18% of the small-oscillation period across the rod lengths.Source: §D.3.2 · p. 22

D.3.3 THE DATA SPLIT PROTOCOL

The appendix specifies episode-level data splits, frozen-probe evaluations, and an actuated-pendulum planning protocol using reachable held-out goals. It also describes how candidate actions are optimized and how success is scored.

D.3.3 THE DATA SPLIT PROTOCOL

  • Episodes are split 70/15/15 with no frames shared across splits, stratified by design cell; the conservative-pendulum data uses only the Δt = 0.1 s slice.Training clips use stride 1 from each training episode’s first 80 frames; the split uses seed 42.Source: §D.3.3 · p. 22

D.4.1 THE PROBES

  • Physics predictions are probed for θ and ω with frozen readouts; the lead comparison uses a k-NN probe fitted at K = 1.Angle is represented as (sin θ, cos θ) and recombined with atan2 to produce wrapped angular error.Source: §D.4.1 · pp. 22–23

D.4.2 PLANNING

  • Planning uses reachable actuated-pendulum goals sampled from held-out trajectories 30 frames after the start, with the same goals shared across models.The protocol uses a 30-step initial horizon, bounded actions held for three frames, and CEM with 300 samples, 30 iterations, and 30 elites.Source: §D.4.2 · pp. 23–24
  • Each model plans in its own latent coordinates with CEM and MPC, without future observations or ground-truth state entering the rollout or objective.Replanning re-encodes only frames already observed; success is scored at the goal timestep against separate angle and velocity tolerances.Source: §D.4.2 · pp. 23–24

E SUPPLEMENTARY EXPERIMENTAL RESULTS

  • The supplementary-results section introduces general LeHami properties and then reports environment-specific results.The section uses results from one environment to illustrate general properties before discussing both studied environments.Source: E SUPPLEMENTARY EXPERIMENTAL RESULTS · p. 24

E.1 LEHAMI REPLACES MANUAL DESIGN CHOICES WITH STRUCTURAL PRIORS AND LOWER COMPUTE

LeHami uses an explicit position–momentum split to represent motion direction structurally, rather than relying on an unstructured predictor to infer velocity from history. On the conservative pendulum, it improves prediction at matched training budgets while using less compute and fewer parameters.

Velocity representation

  • A tied W=3 temporal encoder is invariant to reversing frame order, so it maps (θ, +ω) and (θ, −ω) to the same latent and cannot encode velocity sign.Absolute velocity remains decodable at R2 = 0.92, while signed velocity is at the predict-the-mean floor.Source: §E.1 · p. 25
  • Simply widening the temporal window does not ensure velocity information: the standard JEPA objective permits sign-blind representations that still predict future embeddings.This leaves directional information to implicit architectural choices and exposes the failure only in single-latent state readouts.Source: §E.1 · pp. 24–25
  • LeHami encodes momentum as p = SG(q), an odd temporal finite difference that preserves direction without learning asymmetric temporal-kernel weights.The angle is predominantly decoded from q, while the velocity signal comes almost entirely from p.Source: §E.1 · p. 25

Compute and parameter efficiency

  • LeHami’s error is 1.45–4.2× lower on θ and 1.4–2.3× lower on ω at matched training budgets; its 0.75 PFLOP checkpoint beats LeWM’s best within 6 PFLOP.The comparison is from one seed per model, and LeHami’s total parameter count is 3.03M against LeWM’s 4.44M.Source: §E.1 · pp. 25–26, Table 4 · p. 26

E.2 CONSERVATIVE PENDULUM

On the conservative pendulum, LeHami organizes rollouts along learned-energy level sets and avoids the latent-volume contraction seen in LeWM. Its velocity representation also remains strong across probes and encoder choices.

Latent geometry and volume

  • LeHami rollouts stay on low-dimensional closed sets stratified by learned Hϕ, paralleling the distinct energy-level curves of the physical pendulum.The conservative rollout traces closed curves rather than drifting across projected latent planes.Source: §E.2 · p. 26, Figure 9 · p. 27, Figure 10 · p. 27

Physics prediction

  • The velocity advantage persists across encoder families: LeHami’s anchor ω R2 is 0.929 with ViT and 0.939 with ConvNet, versus −0.031 and −0.018 for the baseline.This encoder-independence check uses a single-rod dataset.Source: §E.2 · p. 27

Pretrained visual features

  • With a frozen DINOv2 encoder, both DINO-LeHami and DINO-WM reduce angle and velocity MAE, while LeHami retains its velocity advantage.For K larger than 60, both models begin to deviate significantly.Source: §E.2 · p. 28, Figure 12 · p. 29

Physics prediction

  • The 25-NN probe decodes ω at R2 = 0.975 from LeHami’s phase space, while the ridge probe reaches only R2 = 0.45.The source attributes the gap to velocity being encoded non-linearly in (q, p).Source: §E.2 · p. 27, Table 7 · p. 29

Latent geometry and volume

  • LeWM contracts under both full-map and newest-block Jacobian measures, while LeHami’s log-volume factor stays at 0 for every tested state.The comparison indicates that LeWM’s contraction is not an artifact of choosing one Jacobian for its three-embedding context.Source: §E.2 · p. 29

E.3 ACTUATED PENDULUM

On actuated-pendulum planning, LeHami improves directional-sign success over coasting at lower compute, but positional goal-reaching limits its joint success. The section also examines latent energy under torque, regularizer placement, rollout predictions, and the information carried by latent coordinates.

Latent geometry under actuation

  • Torque moves actuated-pendulum trajectories across learned-energy levels; symplecticity constrains the map, not the value of Hϕ.Unlike the conservative case, the action does work and the trajectories do not remain nested in disjoint energy shells.Source: §E.3 · p. 29, Figure 14 · p. 30

Regularizer placement

  • Applying SIGReg to q alone reaches the stated q-mode variance target, whereas the joint statistic can be satisfied by suppressing p and inflating q.The momentum block remains constrained because p is estimated as a derivative of q.Source: §E.3 · pp. 29–30

Physics prediction in rollout

  • LeHami’s velocity MAE is lower at small rollout steps, but performance becomes on par with LeWM after K ∼ 10.LeWM has lower angle MAE at smaller steps, with the gap overlapping after K ∼ 40.Source: §E.3 · p. 31, Figure 15 · p. 32

Temporal-window effects

  • A tied W=3 encoder improves R2(|ω|) but lowers R2(ω); overlapping windows also attenuate the predictor’s frame-to-frame directional signal by 34%.The symmetric encoder maps opposite velocity directions to the same latent, while untied weights can distinguish forward from backward motion.Source: §E.3 · pp. 31–32, Table 9 · p. 32
  • LeHami’s odd Savitzky–Golay derivative supplies sign-aware momentum from q, rather than relying on learned asymmetric temporal weights.The goal velocities are not balanced in sign, but the reported difference is insufficient to support a claim that goal distribution favors performance by absolute value.Source: §E.3 · pp. 32–33

Planning results

  • On the sign clause, only LeHami (0.800) and untied LeWM W=3 (0.962) beat coasting (0.680); LeHami does so at 0.59× compute (8.05 against 13.65 TFLOP).Source: §E.3 · p. 33
  • At R = 9, LeHami reaches the goal angle on 0.48 of plans, below tied and untied LeWM W=3 at 0.63 and 0.70, respectively.The paper identifies positional goal-reaching, rather than velocity direction, as the bottleneck for LeHami’s joint success on this task.Source: §E.3 · p. 33

Physical-parameter prediction

  • Both models carry rod length and mass through rollout: 25-NN gives R2 ≥ 0.997 at K = 76, while ridge-readout drift is greater for LeHami.The paper reports that assigning the ridge mass prediction to the nearest supported mass is correct on 99.3% of episodes for both models.Source: §E.3 · p. 34, Table 11 · p. 35

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