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

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

cs.AIOct 13, 2026

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

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

Organizations: Shanghai Jiaotong University

Abstract

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.

Figures & tables

Figure 1: Overview of the proposed ReStep framework. Top: A VAE-based steady-state predictor estimates plausible outcomes, which drives the Anchored Diffusion model to generate complete trajectories. Bottom: Traditional autoregressive models. Our approach mitigates this issue by conditioning trajectory generation on both initial and final constraints. The right-hand bar chart compares the Final State MSE of our method with that of autoregressive methods.

Figure 2: Model details. Left: VAE-based steady-state predictor. Here, $x_{T}$ denote the positions at the final state, while $x_{0}$ and $\alpha$ represent the positions at the initial state and physical attributes. Right: Denoising process. The diffusion-based UNet progressively refines the noisy trajectory $s^{0}$ , where $s_{0}$ and $s_{T}$ correspond to the initial and final state conditions. $M$ denotes the number of particles and $T$ the length of the trajectory in time. This integration of initial and final state conditions enables trajectory completion.

Panel of Figure 3 — Figure 3: MSE comparisons across three FlexiFall scenarios between predicted and ground-truth trajectories. Our method consistently achieves the lowest error across all time steps.

Panel of Figure 4 — Figure 4: Prediction showcases on the FlexiFall dataset . Left: RigidFall scene. Right: Topple scene.

Scene	#Seq	Horizon	#Par	#Obj
FlexiFall
RigidFall	10000	150	192	3
Topple	10000	70–150	256	4
Drape	5000	120	800	3
Physion
Collide	2150	151–452	2383	5

Table 1: Datasets. We report dataset volume, horizon length, average number of particles, and average number of objects, respectively.

Method	$t=50$					$t=100$
Method	Collide	Contain	Drop	Dominoes	Roll	Collide	Contain	Drop	Dominoes	Roll
EGNO ( Xu et al., 2024 )	1.027	0.111	0.085	0.011	0.444	1.703	0.247	0.107	0.029	0.604
GMN ( Huang et al., 2022 )	0.981	0.131	0.077	0.012	0.531	1.592	0.331	0.120	–	0.971
SGNN ( Han et al., 2022a )	1.009	0.128	0.066	0.012	0.377	1.365	0.310	0.091	0.031	0.616
LatentFlow ( Boduljak et al., 2026 )	0.105	0.308	0.049	0.005	0.084	0.261	0.454	–	0.014	0.123
Ours	0.055	0.082	0.011	0.001	0.058	0.105	0.163	0.019	0.002	0.091

Table 2: Rollout MSE on Physion for

t=50

and

t=100

. For each scenario, the best-performing method is highlighted in bold, while ‘–’ indicates that the method failed to produce a result at the corresponding horizon.

Panel of Figure 5 — Figure 5: Comparison of rendered results. Compared with Deformable 3DGS, our method illustrates that more accurate trajectory predictions for both rigid and non-rigid objects.

Method	Fric.	Rest.	Mass
InvGraphics ( Jaques et al., 2019 )	0.1992	0.1998	0.1719
Ours	0.0889	0.0784	0.0796

Table 3: MSE of physical parameter estimation given visual observations. We assess friction, restitution, and mass on the Topple benchmark.

Method	Topple			RigidFall			Drape
Method	SSIM↑	PSNR↑	LPIPS↓	SSIM↑	PSNR↑	LPIPS↓	SSIM↑	PSNR↑	LPIPS↓
In-Domain
DeformGS ( Yang et al., 2024 )	0.9846	43.2263	0.0156	0.9784	35.1682	0.0227	0.9859	41.3140	0.0164
Ours	0.9837	41.9273	0.0171	0.9770	33.3704	0.0215	0.9845	39.0993	0.0157
Out-of-Domain
DeformGS ( Yang et al., 2024 )	0.9662	27.4325	0.0551	0.9707	29.0290	0.0335	0.9796	35.2923	0.0226

Table 4: Comparison of visual dynamics across three physical scenes. Metrics are reported for both in-domain (seen during training) and out-domain (unseen future) time steps. Our model achieves notably lower errors on out-of-distribution time steps, indicating stronger generalization.

Panel of Figure 6 — Figure 6: Prediction diversity of the steady-state predictor (a) and ablation results (b).

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Module	Input Dim	Output Dim	Architecture Details
Encoder $\mathcal{E}_{\theta}(\cdot)$	$N\times(3+3)$	$z$	• Position encoder: MLP $(3\rightarrow 128\rightarrow 128)$ • Velocity encoder: MLP $(3\rightarrow 128\rightarrow 128)$ • Feature concatenation to form per-particle embeddings • $L=3$ graph message-passing blocks with $k=16$ neighbors • Linear projection: Linear $(256\rightarrow z)$
Posterior Network $\mathcal{Q}_{\theta}(\cdot)$	$z$	$\mu_{q},\,\log\sigma_{q}^{2}\in\mathbb{R}^{z}$	• MLP $(z\rightarrow 512\rightarrow 2z)$ with normalization • Output split into mean and log-variance
Prior Network $\mathcal{P}_{\theta}(\cdot)$	$z$	$\mu_{p},\,\log\sigma_{p}^{2}\in\mathbb{R}^{z}$	• MLP $(z\rightarrow 512\rightarrow 2z)$ with normalization • Output split into mean and log-variance
Frame Predictor $\mathcal{F}_{\theta}(\cdot)$	$z+z$	$z$	• MLP $(2z\rightarrow 512\rightarrow z)$ with normalization • Predicts the next latent particle state
Decoder $\mathcal{D}_{\theta}(\cdot)$	$N\times(z+3)$	$N\times 3$	• Node encoder: MLP $((z+3)\rightarrow 256\rightarrow 256)$ • $L=3$ graph message-passing blocks with $k=16$ neighbors • Output MLP $(256\rightarrow 256\rightarrow 3)$

Appendix

Table 5: Network architectures and hyperparameters for Stage 1: Stochastic Goal State Prediction. All MLP layers use ReLU activations unless otherwise specified.

Module	Input Dim	Output Dim	Architecture Details
Convolution Block $C_{\theta}(\cdot,\cdot)$	$(T,M,d_{in})$	$(T,M,d_{out})$	• Conv1d $((T,M,d_{in})\rightarrow(T,M,d_{out}))$ , SiLU • Feature fusion via element-wise addition • Conv1d $((T,M,d_{out})\rightarrow(T,M,d_{out}))$ , SiLU
Attention Block $A_{\theta}(\cdot,\cdot)$	$(T,M,d)$	$(T,M,d)$	• SelfAttention $((T,M,d)\rightarrow(T,M,d))$ • CrossAttention $((T,M,d)\rightarrow(T,M,d))$ • GeGLU $((T,M,d)\rightarrow(T,M,4d)\rightarrow(T,M,d))$

Appendix

Table 6: Network architectures and hyperparameters for Stage 2: Anchored Diffusion. We use the second UNet as an example, which applies convolution along the particle dimension.

Nominal confidence $p$	Coverage	Threshold
0.50	0.3400	0.004806
0.80	0.7867	0.007224
0.90	0.8800	0.009154
0.95	0.9067	0.011467

Appendix

Table 7: Calibration of the steady-state predictor on the Dominoes scene.

Scene	Endpoint condition	$t=50$	$t=100$
Dominoes	VAE	0.0017	0.0026
	GT	0.0012	0.0014
	Random	0.0047	0.0073
Drop	VAE	0.0110	0.0227
	GT	0.0122	0.0174
	Random	0.0400	0.0727

Appendix

Table 8: Trajectory MSE under different endpoint conditions (lower is better). Random endpoints are constructed by adding clipped Gaussian perturbations to the initial condition. The VAE condition uses a stochastic endpoint sample.

Scene	GNN/ $k$ -NN	Transformer
Dominoes	$0.0100\pm 0.0064$	$0.0093\pm 0.0074$
Collide	$0.3137\pm 1.2218$	$0.2946\pm 1.3407$
Drop	$0.0357\pm 0.0974$	$0.0357\pm 0.0928$

Appendix

Table 9: Endpoint MSE (mean

\pm

standard deviation; lower is better) for two endpoint-predictor backbones. These are endpoint prediction errors, not trajectory rollout errors.

Scene	Model	FLOPs (B)	Time (s)	Parameters (M)
Dominoes	SGNN	63.254	1.258	1.551
	ReStep	121.742	0.055	5.36
Roll	SGNN	8.727	1.016	0.993
	ReStep	16.280	0.058	5.36

Appendix

Table 10: Reported normalized inference cost per predicted step. ReStep’s full-trajectory FLOPs and time are divided by trajectory length; SGNN predicts one step at a time. The parameter count is that of the compared models.

Scene	Particles	FLOPs (B)	$k$ -NN (ms)	Forward / total (ms)
Collide	361	14.81	1.53	4.05 / 5.58
Roll	474	19.45	1.49	3.77 / 5.26
Contain	603	24.75	1.52	4.02 / 5.53
Drop	1326	54.48	2.03	6.89 / 8.92
Link	1608	66.10	2.40	8.02 / 10.42
Support	2028	83.42	3.03	9.35 / 12.37

Appendix

Table 11: Stage-1 endpoint-predictor inference for different particle counts (batch size 1). Forward time excludes the separately measured

k

-NN construction; total time includes both.

Endpoint-error group	Traj. Q1	Traj. Q2	Traj. Q3	Traj. Q4
Lowest 25% ( $n=25$ )	23	2	0	0
Highest 25% ( $n=25$ )	0	0	4	21
Highest 10% ( $n=10$ )	0	0	0	10

Appendix

Table 12: Association between endpoint-error groups and trajectory-error quartiles on 100 RigidFall samples. Rows count samples in the lowest and highest endpoint-error quartiles; the remaining 50 samples are omitted.

Scene	Endpoint predictor	Endpoint MSE $\downarrow$
Contain	VAE	$\mathbf{0.3731}\pm 0.5481$
	Deterministic model	$0.4052\pm 0.5824$
Drop	VAE	$\mathbf{0.0357}\pm 0.0974$
	Deterministic model	$0.0484\pm 0.1151$

Appendix

Table 13: Endpoint prediction errors for VAE-based and deterministic predictors on Contain and Drop. Lower mean endpoint MSE is highlighted in bold.

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