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Back to the Future: Reverse-Consequence Modeling for Long-Horizon Particle Dynamics

Leyan Zhang, Nantian He, Jiajun Ma, Yanpeng Zhao, Yunbo Wang

arXiv:2610.18168v1cs.AI

TL;DR

Autoregressive trajectory prediction can accumulate errors over long horizons, while dissipative systems may have more predictable final states than transient paths. ReStep predicts a plausible final state and generates an anchored trajectory; experiments report improved long-horizon accuracy, with an extension to image-based physical inference. Its trajectories remain sensitive to errors in the predicted steady state.

  • Problem

    Autoregressive models accumulate errors over long rollouts, while in dissipative scenes steady states can be more predictable than the transient dynamics leading to them.

    Source: §1 · p. 1

  • Method

    ReStep first uses a stochastic VAE-based predictor to estimate a plausible final state, then generates a trajectory conditioned on the initial and predicted final states.

    Source: §1 · p. 1

  • Results

    ReStep improves long-horizon prediction accuracy and supports latent-parameter recovery and visually faithful dynamics on FlexiFall and Physion tasks.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Integrating ReStep with 3D Gaussian Splatting supports inference of latent physical properties and visual dynamics from images.

    Source: §1 · p. 2

    Errors in the predicted steady state can propagate through diffusion and produce physically implausible trajectories.

    Source: §6 · p. 9

Abstract

from arXiv · show

Accurate prediction of object trajectories is fundamental to mechanical understanding and robotic control. However, autoregressive models inherently suffer from cumulative errors, leading to severe drift in long-horizon rollouts. Observing that in many physical systems with dissipative or convergence-to-rest behavior, the final states are often more predictable than transient dynamics (e.g., a sliding object resting under friction or a pendulum stabilizing at equilibrium), we propose ReStep, a cascaded approach that reverses the conventional forward-prediction paradigm. ReStep first employs a stochastic model to directly predict the final state, then leverages Anchored Diffusion to generate goal-conditioned intermediate trajectories. By conditioning on both ''boundary states'', it effectively constrains dynamics and reduces uncertainty, producing physically plausible and temporally coherent transitions. We further integrate ReStep dynamics as the physical engine within a 3D Gaussian Splatting pipeline, enabling a unified physical-visual framework for intuitive inference from images. Extensive experiments on diverse rigid and non-rigid collision datasets show that ReStep achieves superior long-term accuracy, substantially mitigating error accumulation via reverse-consequence modeling.

1 INTRODUCTION

ReStep reverses long-horizon prediction by estimating a plausible steady state before generating the intervening trajectory. It also integrates this dynamics approach into a 3D Gaussian Splatting pipeline for image-based physical inference.

1 INTRODUCTION

  • Autoregressive prediction can accumulate timestep errors, causing long-horizon trajectories to drift and miss convergence behaviors such as frictional settling.The paper observes that steady states in dissipative scenes can be more predictable than the transient trajectories leading to them.Source: §1 · p. 1
  • A VAE-based stochastic predictor estimates plausible final states, then Future-Anchored Diffusion generates trajectories conditioned on the initial and predicted final states.The approach models a distribution of possible steady states rather than a single deterministic outcome.Source: §1 · pp. 1–2, Figure 1 · p. 2
  • ReStep integrates its dynamics model into a 3D Gaussian Splatting pipeline to infer latent physical properties and generate physically consistent visual dynamics from images.The paper motivates this integration by noting that simulation particles and physical parameters are often unobservable in real-world settings.Source: §1 · p. 2
  • ReStep improves long-horizon prediction accuracy, supports recovery of latent physical parameters from images, and produces visually faithful dynamics on FlexiFall and Physion.The experiments cover particle-based prediction and vision-grounded tasks involving rigid and non-rigid objects.Source: §1 · p. 2

2 RELATED WORK

Prior work models particle dynamics with graph neural networks and uses diffusion methods for conditional generation. The section notes that rollout error accumulation persists despite strong one-step prediction performance.

2 RELATED WORK

  • Particle-dynamics GNNs use neighbor message passing, and some employ equivariant architectures to encode 3D Euclidean symmetries.The cited works report state-of-the-art performance in predicting physical dynamics.Source: §2 · pp. 2–3
  • GNN-based dynamics models perform strongly on one-step transitions, but their trajectory rollouts suffer from error accumulation.These models construct graphs from pairwise particle distances and evolve them through repeated message passing.Source: §2 · pp. 2–3
  • Conditional diffusion approaches include classifier guidance, classifier-free guidance, and goal-conditioned trajectory inpainting or conditioning at denoising steps.The passage describes feature concatenation, FiLM, and cross-attention as ways to inject conditions across denoising steps.Source: §2 · p. 3

3 PROBLEM DEFINITION

The problem is to generate particle trajectories from an initial configuration while accounting for uncertainty in physical attributes and interactions. ReStep completes trajectories using both the initial state and a predicted final state.

3 PROBLEM DEFINITION

  • A physical state combines particle positions with attributes such as velocity, friction, mass, and elasticity, and a trajectory is the sequence of states over time.The passage denotes the state at time t as st = (xt, α).Source: §3 · p. 3
  • Instead of predicting each next state from the current one, the proposed formulation conditions trajectory completion on the initial state and a predicted final state.The target conditional distribution is written as p(s1:T | s0), in contrast to the anchored completion formulation.Source: §3 · p. 3
  • Given particle-based trajectories, the task is to predict a plausible final state from the initial state and generate the intermediate states conditioned on both endpoints.The final-state prediction accounts for uncertainty in attributes and interactions.Source: §3 · p. 3

4 METHOD

ReStep reframes long-horizon dynamics as future-anchored trajectory completion: it predicts a plausible final state, then generates a trajectory between the initial and predicted final states. The framework also integrates these dynamics with 3D Gaussian Splatting for physical inference and visual generation.

4 METHOD

  • ReStep constrains long-horizon motion by predicting a future state and completing the trajectory between that state and the initial state.The framework integrates the resulting dynamics into a 3D Gaussian Splatting pipeline.Source: §4 · p. 3

4.1 VARIATIONAL STEADY-STATE PREDICTION

  • A stochastic GNN-based VAE predicts diverse, physically plausible final-state hypotheses from the initial configuration and physical parameters.Its latent distribution represents uncertainty over possible futures, including different resting configurations from small perturbations.Source: §4.1 · pp. 3–4

4.2 ANCHORED DIFFUSION

  • Anchored Diffusion synthesizes a trajectory conditioned on its initial and final states, with iterative denoising connecting the two endpoints.The Double-UNet models temporal evolution and particle interactions, and is designed to handle varying particle counts.Source: §4.2 · pp. 4–5

4.3.1 INVERSE RENDERING FOR PHYSICAL PARAMETER ESTIMATION

  • The 3D Gaussian Splatting pipeline uses reconstructed point clouds from initial and final multi-view images to obtain particle states for physical parameter estimation.A pretrained steady-state predictor serves as a differentiable surrogate; physical parameters are optimized to reduce discrepancy with the observed final state.Source: §4.3 · p. 5, §4.3.1 · p. 5

4.3.2 RESTEP-DRIVEN VISUAL DYNAMICS GENERATION

  • For visual dynamics generation, ReStep predicts a final particle state, synthesizes the intervening trajectory, and applies motion updates to the reconstructed Gaussian scene.The resulting sequences are described as temporally coherent and consistent with the observed initial appearance and physical properties.Source: §4.3.2 · p. 6

5 EXPERIMENTS

ReStep is evaluated on rigid and non-rigid particle dynamics, visual rendering, and analyses of endpoint prediction, component ablations, and generalization. Results report lower trajectory errors, improved visual parameter estimation, and robustness across the tested shifts.

5.1 DATASETS

  • FlexiFall extends RigidFall with a sequential-contact Topple scene and a cloth-draping scene, while Physion covers rigid and soft-body interactions and other physical scenarios.The datasets are described by volume, horizon length, and average particle and object counts.Source: §5.1 · p. 6, Table 1 · p. 6

5.2 MAIN RESULTS ON FLEXIFALL; 5.3 MAIN RESULTS ON PHYSION

  • ReStep achieves the lowest MSE across all FlexiFall time steps and outperforms prior approaches on Physion at both t = 50 and t = 100.FlexiFall comparisons cover three scenarios; Physion reports rollout MSE at the two horizons.Source: Figure 3 · p. 6, §5.2 · p. 7, Table 2 · p. 7

5.4 RESULTS ON PHYSICS-GROUNDED RENDERING

  • On Topple, MSE falls from 0.1992 to 0.0889 for friction, 0.1998 to 0.0784 for restitution, and 0.1719 to 0.0796 for mass against Physics-as-Inverse-Graphics.The estimates are inferred from visual observations; the evaluation uses normalized physical attributes.Source: §5.4 · pp. 7–8, Table 3 · p. 7
  • For out-of-domain future time steps, ReStep significantly outperforms Deformable 3DGS and maintains structural consistency in rendered dynamics.The evaluation compares visual scene generation on FlexiFall using PSNR, SSIM, and LPIPS; no metric values are supplied here.Source: §5.4 · p. 8, Table 4 · p. 8

5.5.1 ANALYSIS OF THE STEADY-STATE PREDICTOR

  • In 200 RigidFall samples, higher-starting objects have broader final XY-centroid distributions, while the lowest object has a more concentrated distribution.The endpoint predictor’s calibration is close to nominal at high confidence, with some under-coverage at lower confidence.Source: §5.5.1 · p. 8

5.5.2 ABLATION STUDIES

  • On RigidFall, removing steady-state prediction or anchored diffusion increases long-horizon error, especially when steady-state prediction is removed.The comparison also reports lower mean endpoint MSE for the VAE than the deterministic predictor on Contain and Drop.Source: §5.5.2 · p. 9

5.5.3 OUT-OF-DISTRIBUTION GENERALIZATION

  • On Topple composition shifts, steady-state MSE changes from 0.0036±0.0020 to 0.0035±0.0019, while trajectory MSE changes from 0.0010 ± 0.0009 to 0.0019 ± 0.0013.Objects are subsampled from multi-object scenes while interaction rules are preserved.Source: §5.5.3 · p. 9
  • On Dominoes, average trajectory MSE rises from 0.0035 ± 0.0020 in-distribution to 0.0055 ± 0.0035 when trajectories are interpolated to approximately twice their original length.This tests generalization across temporal lengths.Source: §5.5.3 · p. 9

6 CONCLUSIONS AND LIMITATIONS

ReStep conditions trajectory generation on initial and inferred final states, mitigating error accumulation and improving long-horizon stability compared with autoregressive methods. The paper also extends the framework to visually grounded tasks and describes how diffusion-generated motion updates Gaussian scene representations.

6 CONCLUSIONS AND LIMITATIONS

  • The predicted steady state can be erroneous, and those errors can propagate through diffusion to produce physically implausible trajectories.The authors suggest rejection sampling or stronger consistency priors as possible mitigations and future work on more reliable final-state inference.Source: §6 · p. 9

APPENDIX CONTENTS

  • The appendices provide architecture details, additional analyses, implementation information, and code in supplementary materials.The additional analyses cover endpoint conditioning, model backbones, inference cost, particle-count scaling, and failure cases.Source: REPRODUCIBILITY STATEMENT · p. 10

A.1 VAE-BASED PREDICTION

  • VAE-based prediction models future states through latent-space temporal transitions, supporting uncertainty and multimodal dynamics.The encoder maps observations to latent distributions, and the decoder reconstructs states from latent variables.Source: §A.1 · pp. 13–14

A.2 DENOISING DIFFUSION PROBABILISTIC MODELS

  • Diffusion reverses progressive Gaussian noising through iterative denoising, and the paper adapts it to generate trajectories from initial and predicted final states.The approach is used to model diverse motion sequences while pursuing spatial and temporal coherence.Source: §A.2 · p. 14

B DETAILS OF RESTEP-DRIVEN INVERSE RENDERING

  • The inverse-rendering pipeline converts diffusion-generated particle trajectories into Gaussian-scene updates, handling rigid and non-rigid objects separately.The combined formulation is intended to represent both deformable materials and solid objects.Source: B DETAILS OF RESTEP-DRIVEN INVERSE… · p. 14, §B.3 · p. 15

B.1 NON-RIGID OBJECTS

  • For cloth, each Gaussian point receives a weighted displacement from its k nearest cloth particles in the initial configuration.This makes points follow the particle-defined local deformation field while preserving smoothness and spatial coherence; rotations remain unchanged.Source: §B.1 · pp. 14–15

B.2 RIGID OBJECTS

  • For rigid objects, a single per-object rotation and translation align the initial and diffusion-predicted particle clouds, then apply to all Gaussian points.The alignment uses the Kabsch algorithm and preserves object shape.Source: §B.2 · p. 15

B.3 COMBINED UPDATE

  • At each diffusion step, the Gaussian scene combines non-rigid and rigid motions to represent both deformable materials and solid objects.The paper describes this hybrid update as supporting faithful scene motion and accurate reconstruction of both behaviors.Source: §B.3 · p. 15

C ARCHITECTURE DETAILS

The architecture details distinguish a stochastic goal-state prediction stage from a second-stage Anchored Diffusion model. The latter uses a Double-UNet with convolution and attention blocks.

C.1 STAGE 1: STOCHASTIC GOAL STATE PREDICTION

  • Stage 1 represents each particle by position and velocity and uses graph-based modules with k-nearest-neighbor message passing to model local interactions.The passage specifies N particles and spatial dimension d = 3; MLPs use ReLU unless otherwise specified.Source: §C.1 · p. 15

C.2 STAGE 2: ANCHORED DIFFUSION

  • Stage 2 implements Anchored Diffusion with a Double-UNet containing convolution and attention operations.The architecture description corresponds to the hyperparameters summarized in Table 6.Source: §C.2 · p. 15
  • Each attention block uses self-attention, cross-attention for condition injection, and GeGLU gating, with residual connections and group normalization.Convolution blocks use two 1D convolution layers; downsampling uses strided convolution and upsampling uses nearest-neighbor interpolation.Source: §C.2 · p. 15

D.1 CALIBRATION OF THE ENDPOINT PREDICTOR

The endpoint predictor’s uncertainty is assessed through coverage, and endpoint inputs are compared for trajectory generation. A Transformer-style predictor also provides a backbone comparison.

D.1 CALIBRATION OF THE ENDPOINT PREDICTOR

  • 0.8800 coverage at p = 0.90 and 0.9067 at p = 0.95; at p = 0.50, coverage is 0.3400 on Dominoes.Coverage is close to nominal at higher confidence levels, while the lower-confidence result indicates under-coverage in more nonlinear contact-rich cases; the threshold rises monotonically with p.Source: §D.1 · p. 16, Table 7 · p. 16

D.2 SENSITIVITY TO THE ENDPOINT CONDITION

  • The VAE-predicted endpoint outperforms the perturbed endpoint in all reported scenes, while the observed GT endpoint is generally most informative.The ordering is not uniform: the VAE condition slightly improves on GT for Drop at t = 50, while the gap is particularly large on Roll at t = 100.Source: §D.2 · p. 16, Table 8 · p. 16

D.3 ENDPOINT-PREDICTOR BACKBONE

  • The Transformer-style endpoint predictor has comparable or slightly lower mean endpoint MSE than the GNN/k-NN predictor in the evaluated scenes.The authors therefore do not attribute endpoint conditioning’s benefit to the GNN backbone; the GNN remains a locality-based choice for particle interactions.Source: §D.3 · p. 16, Table 9 · p. 17

D.4 INFERENCE COST AND PARTICLE-COUNT SCALING

The inference-cost comparison normalizes ReStep’s full-trajectory computation by trajectory length against SGNN’s one-step prediction. A separate stage-1 profile reports endpoint-predictor timing across particle counts.

D.4 INFERENCE COST AND PARTICLE-COUNT SCALING

  • On two scenes, ReStep uses more FLOPs and parameters but has lower measured wall-clock time than SGNN under the reported normalized inference-cost comparison.ReStep’s full-trajectory FLOPs and time are divided by trajectory length; the comparison reflects evaluated implementations and hardware.Source: §D.4 · p. 17, Table 10 · p. 17
  • Stage-1 endpoint-predictor total time ranges from 5.26 to 12.37 ms as effective particle count ranges from 474 to 2028 in the evaluated scenes.Each forward pass builds a k = 16 neighbor graph six times; total time includes k-NN construction, while the separate construction time is excluded from forward-pass time.Source: §D.4 · p. 17, Table 11 · p. 17
  • The stage-1 timing profile is distinct from the full-model comparison and should not be interpreted as full trajectory-generation latency.Static, non-informative large objects are excluded from the effective-particle counts.Source: §D.4 · p. 17, Table 11 · p. 17

D.5 ENDPOINT ERRORS AND FAILURE CASES

Endpoint-error groups tend to align with trajectory-error quartiles, though this association does not establish causation. The VAE-based predictor also achieves lower mean endpoint MSE than the deterministic predictor on Contain and Drop.

D.5 ENDPOINT ERRORS AND FAILURE CASES

  • 23 of 25 lowest-endpoint-error samples also have the lowest trajectory-error quartile, while 21 of 25 highest-endpoint-error samples fall in the highest quartile.The association identifies endpoint errors as a consequential failure mode but does not establish how much trajectory error they cause.Source: §D.5 · p. 17, Table 12 · p. 18
  • All ten samples above the 90th percentile of endpoint MSE fall in the highest trajectory-error quartile.Their mean endpoint MSE is 0.2688, mean trajectory MSE is 0.1277, and mean trajectory MSE at t = 100 is 0.2488.Source: §D.5 · p. 17, Table 12 · p. 18

D.6 VAE-BASED VERSUS DETERMINISTIC ENDPOINT PREDICTION

  • The VAE lowers mean endpoint MSE from 0.4052 to 0.3731 on Contain and from 0.0484 to 0.0357 on Drop versus the deterministic predictor.These correspond to relative reductions of 7.9% and 26.2%, respectively.Source: Table 13 · p. 18, §D.6 · p. 18
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