Interactive 4D Volumetric Liquid Forecasting under Moving-Solid Interaction

Jongwon Ryu; Sunjae Yoon; Junyeong Kim

cs.AIOct 12, 2026

Interactive 4D Volumetric Liquid Forecasting under Moving-Solid Interaction

Authors: Jongwon Ryu, Sunjae Yoon, Junyeong Kim

Organizations: Department of Artificial Intelligence Chung-Ang University

Abstract

We propose the task of Interactive 4D Volumetric Liquid Forecasting: predicting the spatiotemporal evolution of a future dense 3D liquid velocity field conditioned on how a solid object moves through it. This task provides an upstream physical forecasting primitive for systems that need to reason about motion-conditioned liquid response, including robotic liquid manipulation, digital twins, and rigid-fluid video generation. Existing works have made efforts in predicting flow around static or prescribed solid geometries and synthesizing visually plausible fluid motion. Yet these approaches provide limited coverage for interactive liquid dynamics, as they are not capable of forecasting how a moving solid induces a future full-domain liquid response. In this forecasting scenario, the hydrodynamic effect is condensed near an evolving liquid-solid interface, but the prediction target is a global spatiotemporal rollout. To address this challenge, we propose a two-level recurrent forecasting framework, TIDE/TIDES: TIDE isolates localized interface-driven forecasting, while TIDES adds transport-aware coupling and explicit stabilization to reduce accumulated rollout error. Across adapted neighboring baselines, controlled component ablations, motion/size OOD tests, and long-horizon evaluation, TIDES achieves the best forecasting performance and the ablations identify localized interaction, transport coupling, and stabilization as the key contributors. Together, our task formulation, benchmark, and TIDE/TIDES study establish a controlled foundation for moving-solid-conditioned dense liquid forecasting. The code is publicly available at https://github.com/jongwonryu/TIDE-TIDES.

Figures & tables

Figure 1: Task overview. Left: moving-solid-conditioned forecasting setup, where the current liquid velocity field and the known solid motion sequence are provided to an autoregressive liquid forecaster. Right: the corresponding future liquid velocity rollout.

Figure 2: Method overview. TIDE provides the localized interface-driven core, while TIDES augments it with transport feedback and stabilization modules for autoregressive rollout.

Relative interface distance bin	Mean response energy
$[0.00,0.05)$	1.663763
$[0.05,0.10)$	0.344675
$[0.10,0.20)$	0.104469
$[0.20,0.35)$	0.012967
$[0.35,0.50)$	0.004842
$[0.50,1.00)$	0.003659

Table 1: Distance-to-interface response energy. Lower interface distance corresponds to voxels closer to the evolving interaction boundary.

Model	$l_{\mathrm{roll}}\downarrow$	$traj_{\mathrm{end}}\downarrow$	$\mathrm{near}\downarrow$	$\mathrm{far}\downarrow$	$\Delta_{\mathrm{div}}$
3D ResUNet	0.007585	0.122781	0.005757	0.009553	+0.000000
U-NO	0.006542	0.112111	0.005807	0.005999	-0.000376
GNS	3.767170	5.555210	3.673578	3.957512	+0.000000
Transolver	0.006981	0.142872	0.005712	0.008737	+0.000000
Bulk Only	0.007366	0.121042	0.005771	0.007495	+0.000241
Finite Support	0.006240	0.113406	0.005855	0.006389	-0.000150

Table 2: Main comparison under the same optimization budget and evaluation protocol, aggregated over 5 random seeds. Bold indicates the best performance.

Figure 3: Qualitative comparison of autoregressive velocity rollouts. Each column shows a model and each row shows a rollout step from $t$ to $t{+}16$ . The white contour indicates the moving-solid location,and brighter colors indicating higher velocity magnitude.

Variant	$l_{\mathrm{roll}}\downarrow$	$traj_{\mathrm{end}}\downarrow$	$\mathrm{near}\downarrow$	$\mathrm{far}\downarrow$	$\Delta_{\mathrm{div}}$
Localized(TIDE)	0.006113	0.110904	0.005660	0.006279	-0.000248
w/o projection	0.006943	0.118163	0.005844	0.006985	+0.000000
w/o objpatch	0.006357	0.111277	0.005727	0.006406	+0.000140
with Global Conditioning	0.006542	0.112111	0.006807	0.006599	-0.000076
Finite Support	0.006240	0.113406	0.005855	0.006389	-0.000050

Table 3: Localized-centered principle ablation. Bold indicates the best performance.

Variant	$l_{\mathrm{roll}}\downarrow$	$traj_{\mathrm{end}}\downarrow$	$\mathrm{near}\downarrow$	$\mathrm{far}\downarrow$	$\Delta_{\mathrm{div}}$
TIDES	0.005746	0.104397	0.005581	0.005826	-0.057067
w/o dual feedback	0.005906	0.153737	0.005671	0.005973	-0.009762
w/o latent correction	0.005750	0.109175	0.005642	0.005975	-0.008878
w/o output correction	0.006017	0.108425	0.005662	0.006139	-0.002105
w/o both corrections	0.005825	0.162747	0.005690	0.005837	-0.009326

Table 4: Full-model component ablation. Bold indicates the best performance.

Variant	$l_{\mathrm{roll}}\downarrow$	$traj_{\mathrm{end}}\downarrow$	$\mathrm{near}\downarrow$	$\mathrm{far}\downarrow$	$\Delta_{\mathrm{div}}$
Full Loss	0.005746	0.104397	0.005581	0.005826	-0.057067
w/o $L_{\mathrm{traj}}$	0.005801	0.110317	0.005768	0.005951	-0.030720
w/o $L_{\mathrm{int}}$	0.005867	0.105963	0.005638	0.005903	-0.045270
w/o $L_{\mathrm{roll}}$	0.006907	0.105727	0.006012	0.007216	-0.041455

Table 5: Single-loss ablation of the final Full objective on the Full Stabilized model. Each row removes one loss term while keeping the architecture fixed. Bold indicates the best performance.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Input	Conditioning	Output	Evaluation
3D ResUNet	voxel grid	global solid channels	voxel velocity	voxel
U-NO	voxel grid	global solid channels	voxel velocity	voxel
Transolver	voxel grid	global solid-conditioning grid	voxel velocity	voxel
GNS	particle state	kinematic rigid proxies	particle state	voxelized via P2G

Appendix

Table 6: Implementation details for adapted external baselines under the dense-liquid, simplified-input protocol.

Method	Rollout-8 latency	Relative speed vs. DualSPHysics
DualSPHysics (reference simulator)	6.46 s	1.0 $\times$
3D ResUNet	130.2 ms	49.6 $\times$
U-NO	201.9 ms	32.0 $\times$
GNS	77.4 ms	83.5 $\times$
Transolver	313.5 ms	20.6 $\times$
TIDE	202.0 ms	32.0 $\times$

Appendix

Table 7: Controlled rollout-8 latency comparison. Learned-model times are measured for a single 8-step query under the shared benchmark protocol. The DualSPHysics reference is the median simulation runtime per physical second across two representative benchmark logs.

Loss term	Symbol	Weight
Velocity reconstruction	$\lambda_{\mathrm{vel}}$	1.0
Gradient / vorticity consistency	$\lambda_{\mathrm{grad}}$	0.2
Divergence regularization	$\lambda_{\mathrm{div}}$	0.1
Trajectory supervision	$\lambda_{\mathrm{traj}}$	0.5
Interface consistency	$\lambda_{\mathrm{int}}$	0.2
Rollout consistency	$\lambda_{\mathrm{roll}}$	0.5

Appendix

Table 8: Loss terms used in the dense-simplified implementation.

Model	$l_{\mathrm{roll}}\downarrow$	$\mathrm{traj\_end}\downarrow$	$\mathrm{near}\downarrow$	$\mathrm{far}\downarrow$	$\Delta_{\mathrm{div}}$
3D ResUNet	$0.007585$ $\pm 0.0011$	$0.122781$ $\pm 0.1412$	$0.005757$ $\pm 0.0015$	$0.009553$ $\pm 0.0031$	$+0.000000$ $\pm 0.0007$
U-NO	$0.006542$ $\pm 0.0005$	$0.112111$ $\pm 0.1679$	$0.005807$ $\pm 0.0020$	$0.005999$ $\pm 0.0021$	$-0.000376$ $\pm 0.0008$
GNS	$3.767170$ $\pm 0.0286$	$5.555210$ $\pm 1.8474$	$3.673578$ $\pm 0.0286$	$3.957512$ $\pm 0.0286$	$+0.000000$ $\pm 0.0032$
Transolver	$0.006981$ $\pm 0.0007$	$0.142872$ $\pm 0.0575$	$0.005712$ $\pm 0.0024$	$0.008737$ $\pm 0.0028$	$+0.000000$ $\pm 0.0007$
Bulk Only	$0.007366$ $\pm 0.0007$	$0.121042$ $\pm 0.0188$	$0.005771$ $\pm 0.0018$	$0.007495$ $\pm 0.0020$	$+0.000241$ $\pm 0.0007$
Finite Support	$0.006240$ $\pm 0.0009$	$0.113406$ $\pm 0.0413$	$0.005855$ $\pm 0.0019$	$0.006389$ $\pm 0.0022$	$-0.000150$ $\pm 0.0007$

Appendix

Table 9: Seed-stability version of Table 2 . The mean values match the main-text table, and the appendix additionally reports mean

\pm

standard deviation across 5 random seeds. Lower is better.

Model	$l_{\mathrm{roll}}\downarrow$	$traj_{\mathrm{end}}\downarrow$	$\mathrm{near}\downarrow$	$\mathrm{far}\downarrow$	$\Delta_{\mathrm{div}}$
3D ResUNet	0.008630	0.135718	0.006614	0.010646	+0.000000
U-NO	0.007816	0.140968	0.006769	0.007073	-0.000371
Transolver	0.008277	0.211586	0.006864	0.010068	+0.000000
GNS	4.150108	8.790151	4.150108	4.150108	+0.000000
TIDE	0.007153	0.146672	0.006494	0.007417	-0.000224
TIDES	0.006771	0.132804	0.006457	0.006950	-0.056732

Appendix

Table 10: OOD results on motion/trajectory OOD. Lower is better.

Model	$l_{\mathrm{roll}}\downarrow$	$traj_{\mathrm{end}}\downarrow$	$\mathrm{near}\downarrow$	$\mathrm{far}\downarrow$	$\Delta_{\mathrm{div}}$
3D ResUNet	0.021598	2.264266	0.011117	0.029733	+0.000000
U-NO	0.005987	0.698487	0.005981	0.006846	-0.000364
Transolver	0.009305	0.843196	0.009604	0.121690	+0.000000
GNS	3.107471	4.115864	3.085135	3.117987	+0.000000
Bulk Only	0.006530	0.773276	0.005943	0.006590	+0.000214
Finite Support	0.006786	0.715258	0.006279	0.006669	-0.000159

Appendix

Table 11: OOD results on official size OOD, where held-out sphere and cone cases use doubled radius (2x) and held-out cube cases use doubled edge length (2x) at test time. Lower is better.

Step	Model	$l_{\mathrm{roll}}\downarrow$	$\mathrm{near}\downarrow$	$\mathrm{far}\downarrow$	$\Delta_{\mathrm{div}}$
16	3D ResUNet	0.009947	0.006339	0.012917	+0.000000
	U-NO	0.006632	0.005664	0.005562	-0.000296
	GNS	8.067133	8.065323	8.277538	+0.000000
	Transolver	0.009115	0.006001	0.011649	+0.000000
	TIDE	0.006366	0.006130	0.006269	-0.000115
	TIDES	0.005964	0.005911	0.005869	-0.048668

Appendix

Table 12: Long-horizon forecasting on the in-distribution test split after training with horizon 8. Rows are grouped by evaluation rollout step. Lower is better.

Figure 4: Long-horizon velocity error growth on the test split.

Figure 5: Representative size-OOD dataset examples used for object-scale generalization evaluation.

Model	Params.	$l_{\mathrm{roll}}$	$traj_{\mathrm{end}}$	$l_{\mathrm{roll}}^{H=24}$
U-NO, global input	338.0M	0.00570	0.132	0.00683
U-NO, localized input	337.9M	0.00548	0.124	0.00660
TIDE	338.0M	0.00530	0.112	0.00638
TIDES	338.3M	0.00502	0.104	0.00598

Appendix

Table 13: Additional parameter-matched control under a common extended optimization schedule. Values are means over five paired training seeds. The final column reports rollout error at evaluation horizon 24. Lower forecasting error is better.

Model	Reference	Shared overwrite	Shared physical losses
U-NO	0.00570	0.00565	0.00549
TIDE	0.00530	0.00527	0.00520
TIDES	0.00502	0.00499	0.00492

Appendix

Table 14: Rollout error under shared boundary enforcement and physical-loss controls. The reference column uses the additional extended-schedule runs.

Evaluation target or support	Localized U-NO	TIDE	TIDES
Nearest-support dense target	0.00548	0.00530	0.00502
SPH-kernel interpolation	0.00542	0.00524	0.00497
Kernel-supported voxels only	0.00535	0.00517	0.00490
Smoothed dense target	0.00527	0.00510	0.00483

Appendix

Table 15: Target-construction sensitivity using fixed predictions. Lower rollout error is better.

RMS error (voxels)	$l_{\mathrm{roll}}$	Interface error	$traj_{\mathrm{end}}$	Increase
0.00	0.00502	0.0254	0.104	0.0%
0.16	0.00504	0.0256	0.105	0.4%
0.40	0.00510	0.0262	0.108	1.6%
0.80	0.00528	0.0280	0.114	5.2%
1.60	0.00569	0.0327	0.129	13.3%

Appendix

Table 16: TIDES sensitivity to smooth perturbations of future solid-state conditioning. The final column gives the relative increase in rollout error over unperturbed conditioning.

Model	$t+1$	$t+8$	$t+16$	$t+24$
U-NO	0.0393 / 0.00594	0.0256 / 0.00660	0.0263 / 0.00595	0.0408 / 0.00650
TIDE	0.0384 / 0.00590	0.0250 / 0.00635	0.0260 / 0.00574	0.0407 / 0.00625
TIDES	0.0380 / 0.00585	0.0245 / 0.00575	0.0248 / 0.00520	0.0385 / 0.00565

Appendix

Table 17: Selected free-running interface/far-field errors. Each cell reports interface error / far-field error under the geometry-derived diagnostic masks.

Model	$E_{\mathrm{change}}$	$E_{\mathrm{div}}$	$D_{\mathrm{abs}}$	$E_{\mathrm{vort}}$	$E_{\mathrm{energy}}$
U-NO	0.004086	0.003054	0.002511	0.002359	0.000573
TIDE	0.004169	0.004082	0.002517	0.002670	0.000610
TIDES	0.003960	0.003020	0.001580	0.002330	0.000565

Appendix

Table 18: Additional diagnostics averaged over free-running offsets 1–24. Lower values are better.

Model	$l_{\mathrm{roll}}^{H=48}$	$l_{\mathrm{roll}}^{H=64}$	$D_{\mathrm{abs},48}$	$D_{\mathrm{abs},64}$
U-NO	0.00755	0.00815	0.0033	0.0038
TIDE	0.00713	0.00747	0.0031	0.0033
TIDES	0.00642	0.00665	0.0019	0.0021

Appendix

Table 19: Additional long-horizon evaluation. Rollout errors are reported at evaluation horizons 48 and 64; divergence columns report absolute predicted divergence at the corresponding final offsets.

Model	Interface	Far field
U-NO	0.0050 / 0.0140	0.0026 / 0.0065
TIDE	0.0049 / 0.0155	0.0022 / 0.0061
TIDES	0.0031 / 0.0090	0.0015 / 0.0040

Appendix

Table 20: Regional absolute-divergence statistics at offset 64. Each cell reports mean / 95th percentile.

Model	Offset 24	Offset 48	Offset 64
U-NO	2.5%	4.2%	5.5%
TIDE	2.3%	3.8%	4.8%
TIDES	1.4%	2.1%	2.8%

Appendix

Table 21: Relative volume drift from shared advected-tracer reconstruction. Lower drift is better.

Model	Sphere	Cube	Cone	Low rot.	Mid rot.	High rot.
U-NO	0.00555	0.00572	0.00583	0.00558	0.00570	0.00584
TIDE	0.00518	0.00531	0.00541	0.00520	0.00530	0.00543
TIDES	0.00490	0.00502	0.00514	0.00491	0.00501	0.00514

Appendix

Table 22: Additional rollout-error breakdown by object family and solid rotation group. Lower is better.

Method	Runtime / physical second	Error
Coarse SPH	0.3015 s	0.00725
TIDES	0.3029 s	0.00502
Full DualSPHysics	6.46 s median	Reference target

Appendix

Table 23: Accuracy comparison at matched runtime. Error is averaged over the matched evaluation set against the high-fidelity target.

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Abstract

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