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Interactive 4D Volumetric Liquid Forecasting under Moving-Solid Interaction
Jongwon Ryu, Sunjae Yoon, Junyeong Kim
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Abstract
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 this https URL.
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- Source: https://arxiv.org/abs/2610.16167v1
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Interactive 4D Volumetric Liquid Forecasting under Moving-Solid Interaction Jongwon Ryu Sunjae Yoon Junyeong Kim â thanks: Corresponding author. Affiliation: Department of Artificial Intelligence Affiliation: Chung-Ang University Email: fbwhddnjs511,sunjaeyoon,junyeongkim@cau.ac.kr 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. 1 Introduction When a solid object moves through liquid, its motion first creates disturbances near the evolving solidâliquid interface, and these local disturbances propagate into a global flow response over time (Kolahdouz et al., 2021; Xiao et al., 2024; Tao et al., 2026; Zhong and Meidani, 2025). We formalize this setting as the task of interactive 4D volumetric liquid forecasting: given an initial liquid state and a future moving-solid trajectory, the goal is to predict the future dense 3D liquid velocity field over the full domain. This task asks whether a model can infer not only where liquid particles move, but also how the full volumetric liquid state evolves as an explicit physical field when a solid object moves, pushes, stirs, or passes through the liquid. This capability provides an upstream physical forecasting primitive for interactive systems that must reason about motion-conditioned liquid response. In robotic liquid manipulation, a system may need to compare candidate pouring, stirring, scooping, or pushing motions before acting (Midlagajni and Rothkopf, 2025). In digital twins and simulation-driven design, operators may wish to ask how changes in a moving tool, actuator, or trajectory alter the surrounding flow state (Kochkov et al., 2021; Li et al., 2023c). In rigidâfluid video generation and future 4D world models, realistic interaction requires not only moving the rigid object correctly, but also producing coherent wakes, waves, and other liquid responses induced by that motion (Akkerman et al., 2025; Shi et al., 2024; Jia et al., 2025). These settings motivate a reusable forecasting primitive for repeated querying: many candidate solid motions can be evaluated under a shared physical scene. Existing learned flow models address related but incomplete settings. Neural operators and CFD surrogates have made substantial progress in efficient field prediction, operator learning, and geometry-aware surrogate modeling (Li et al., 2020; Kovachki et al., 2023; Rahman et al., 2022; Wen et al., 2022; Tran et al., 2021; Li et al., 2023b). However, these methods typically focus on operator regression, geometry-conditioned prediction, or flow around static or prescribed solid geometries, rather than forecasting the evolving liquid response induced by a moving solid. Volumetric flow enhancement methods learn dense velocity fields, but usually reconstruct or enhance already observed measurements rather than predict future motion-conditioned liquid response (Ferdian et al., 2020; Rutkowski et al., 2021; Suk et al., 2026). Visual rigidâfluid synthesis methods can generate plausible fluid appearance, but they often do not provide an explicit dense physical state that can be reused for rollout, querying, or downstream reasoning (Akkerman et al., 2025; Shi et al., 2024; Jia et al., 2025; Du et al., 2024). Particle-, mesh-, and graph-based learned dynamics capture local interactions in simulator-native representations, but their primary prediction space is not a full-domain dense Eulerian volume (Pfaff et al., 2020; Gaoa et al., 2025). Similarly, scene-flow methods estimate 3D motion fields, but they target scene or point motion rather than physically explicit liquid-state forecasting (Li et al., 2023a; Vedder et al., 2024; Liang et al., 2025). Together, these limitations open the problem of forecasting full-domain liquid response under time-varying solid interaction. The key challenge is local-to-global forecasting: the solid directly affects the liquid near the moving solidâliquid interface, but this local disturbance propagates through the full domain and must be predicted as a dense Eulerian velocity rollout. The model must therefore forecast the future dense Eulerian velocity field, while remaining stable under autoregressive rollout and distribution shifts in motion, shape, and horizon (Koehler et al., 2024; Zhong and Meidani, 2025; Shaffer et al., 2026). To address these challenges, we introduce a benchmark for moving-solid-conditioned dense liquid forecasting and propose a recurrent forecasting framework with two levels. TIDE (Temporal Interface-Driven liquid field Estimator) isolates the localized interface-driven forecasting principle: solid-induced updates should be generated where the interaction occurs and then propagated through rollout. TIDES retains this localized core and augments it with transport-aware coupling and explicit stabilization for robust long-horizon forecasting. Our benchmark and shared evaluation protocol are designed for this task. Since no standard benchmark or exact task-matched baseline exists for moving-solid-conditioned dense liquid forecasting, we compare against task-necessity baselines, adapted neighboring baselines, and controlled within-family ablations under a shared forecasting protocol, following the broader emphasis on rollout, physical fidelity, OOD generalization, and benchmarked scientific forecasting (Yang et al., 2025; Hu et al., 2026). Our experiments reveal a clear modeling hierarchy. Localized interface-driven interaction provides the forecasting core, while transport-aware coupling and explicit stabilization improve long-horizon rollout behavior. Under the shared benchmark, TIDES achieves consistent gains in aggregate accuracy, transport fidelity, and robustness across controlled ablations, adapted neighboring baselines, motion/size OOD tests, and long-horizon evaluation shifts. 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. 2 Related Work Neural fluid surrogates and neural operators. A large body of work studies learned surrogates for PDEs and fluid dynamics, especially through neural operator frameworks such as FNO, U-NO, U-FNO, and Geo-FNO (Li et al., 2020; Kovachki et al., 2023; Rahman et al., 2022; Wen et al., 2022; Li et al., 2023b). These models have shown strong performance in operator regression, multiscale prediction, and geometry-aware field modeling (Li et al., 2023c; Wu et al., 2024; Wen et al., 2025; Rabeh et al., 2025). However, they are typically designed for operator regression or geometry-conditioned surrogate prediction. In contrast, our setting requires forecasting a dense liquid rollout under time-varying solid motion, where objectâfluid exchange occurs through an evolving interaction interface. Volumetric flow prediction and enhancement. Another related line of work learns 3D spatiotemporal velocity fields in scientific and medical settings. For example, 4DFlowNet and related methods use CFD-assisted supervision to super-resolve or denoise 4D flow MRI (Ferdian et al., 2020; Rutkowski et al., 2021). These methods show that volumetric velocity fields and local structures such as near-wall behavior can be learned effectively (Rutkowski et al., 2021; Suk et al., 2026). However, they enhance or reconstruct already observed flow measurements, whereas we forecast future liquid response conditioned on changing solid motion and occupancy. Interface-aware fluidâsolid learning and particle/mesh dynamics. Work on fluidâsolid interaction highlights the importance of coupling through boundaries or interfaces rather than uniform global updates (Kolahdouz et al., 2021; Zhong and Meidani, 2025; Xiao et al., 2024). Classical FSI formulations and recent interface-aware learning methods both support this view (Kolahdouz et al., 2021; Tao et al., 2026). In parallel, particle-graph simulators such as GNS and mesh-based learned simulators such as MeshGraphNets provide strong inductive biases for local interaction and transport-consistent dynamics (Pfaff et al., 2020; Midlagajni and Rothkopf, 2025). Our work draws on these insights but targets a different prediction space: dense Eulerian liquid velocity rollout, using Lagrangian transport only as a coupled consistency mechanism rather than as the primary output. Benchmarking scientific forecasting models. Recent benchmarks emphasize evaluating scientific forecasting beyond one-step regression accuracy (Koehler et al., 2024; Hu et al., 2026). APEBench highlights rollout-based and long-horizon evaluation for PDE emulators (Koehler et al., 2024), while ML4CFD stresses accuracy, physical fidelity, out-of-distribution generalization, and efficiency (Yagoubi et al., 2025). This is especially relevant for interactive solidâfluid forecasting, where local interface errors can accumulate into transport or stability failures over autoregressive rollout. Since no exact task-matched benchmark exists for our setting, we evaluate adapted neighboring baselines and controlled within-family ablations under a shared protocol to isolate necessary modeling ingredients. Figure 2: Method overview. TIDE provides the localized interface-driven core, while TIDES augments it with transport feedback and stabilization modules for autoregressive rollout. 3 Method 3.1 Problem Setting We study moving-solid-conditioned interactive 4D volumetric liquid forecasting. Given the current Eulerian liquid velocity field utu_t and a known future solid-state sequence, the model predicts a dense Eulerian liquid-velocity rollout over horizon T: u^t+1:t+T=fΞ(ut,St+1:t+T),SÏ=(ÏÏs,vÏs). u_t+1:t+T=f_Ξ(u_t,S_t+1:t+T), S_Ï=( _Ï^s,v_Ï^s). Here utââ3ĂDĂHĂWu_t ^3Ă DĂ HĂ W is the current 3D liquid velocity field, ÏÏs _Ï^s is the signed-distance field (SDF) of the moving solid, and vÏsv_Ï^s is the solid velocity field rendered on the same voxel grid. SDF-based geometric representations are widely used in geometry-aware neural operators and 3D flow surrogates because they provide a continuous encoding of object boundaries and near-surface structure (Rabeh et al., 2025; Li et al., 2023c). The future solid sequence is treated as known exogenous conditioning; the model predicts only the liquid response. This matches repeated-query settings where candidate solid motions are provided and the goal is to forecast their induced liquid dynamics, which is closely related to rigid-body FSI settings where solid motion acts as a control signal for the surrounding fluid (Zhong and Meidani, 2025). Rather than treating many geometry cues as independent inputs, we derive them deterministically from the future solid state. Specifically, from (ÏÏs,vÏs)( _Ï^s,v_Ï^s) we compute solid occupancy, interface masks, surface normals, and boundary-aligned velocity cues: oÏs=[ÏÏsâ€0],mÏâ=Edge(oÏs),nÏ=âÏÏs.o_Ï^s=1[ _Ï^s†0], m_Ï^â=Edge(o_Ï^s), n_Ï=â _Ï^s. We collect these solid-derived fields into an interface representation âÏ=âĄ(ÏÏs,vÏs)=oÏs,mÏâ,nÏ,vÏs.I_Ï=D( _Ï^s,v_Ï^s)=\o_Ï^s,m_Ï^â,n_Ï,v_Ï^s\. In implementation, these fields are voxel-aligned with the liquid grid and are encoded by lightweight 3D convolutional encoders before being used by the interaction modules. The central difficulty is local-to-global: solid-induced changes originate near the evolving solidâliquid interface, but the target is a stable full-domain Eulerian rollout. This interface-mediated coupling is a central challenge in classical and learned fluidâstructure interaction (Kolahdouz et al., 2021; Tao et al., 2026). 3.2 TIDE: Localized Interface-Driven Core TIDE isolates the core modeling principle of the task: moving-solid effects should be generated near the solidâliquid interface and then propagated through autoregressive rollout, rather than injected everywhere as unstructured global conditioning. This design follows the observation from FSI formulations that objectâfluid exchange is mediated through boundary or interface conditions (Kolahdouz et al., 2021; Zhong and Meidani, 2025). We express this one-step localized forecaster as u^t+1core=Î âĄ(BΞâ(ut)+ÎâuΞintâ(ut,ât+1)), u_t+1^core= \! (B_Ξ(u_t)+ u_Ξ^int(u_t,I_t+1) ), (1) where BΞB_Ξ predicts the bulk volumetric liquid evolution, ÎâuΞint u_Ξ^int predicts a solid-induced residual update conditioned on the next-step interface representation, and Î is a lightweight projection or stabilization operator. The two terms in Eq. (1) play different roles. The bulk path models liquid evolution over the full volume, while the interface path focuses on changes caused by the moving solid. In implementation, the bulk predictor is realized with a compact 3D volumetric backbone, and the interface residual is produced from voxelized interface features using local 3D convolutional processing. This makes TIDE a controlled model for testing whether explicitly localized interface conditioning is necessary, while avoiding additional transport or correction modules used in the full model. 3.3 TIDES: Transport-Aware Stabilized Extension TIDES retains the localized interface-driven core of TIDE, but makes it more robust for autoregressive rollout by adding transport-aware feedback and explicit stabilization. Long-horizon rollout stability is a recurring concern in learned PDE and fluid emulators, where one-step accuracy alone is often insufficient for reliable forecasting (Koehler et al., 2024). It operates in latent space. Let zftz_f^t denote the current fluid latent grid, and let ât+1I_t+1 denote the derived interface representation from the next solid state. TIDES first computes a tentative next fluid latent by combining a bulk fluid update with an interface-mediated residual: z~ft+1=zft+Îâzfbulk+Îâzfint. z_f^t+1=z_f^t+ z_f^bulk+ z_f^int. (2) The bulk update is produced by 3D U-NO-style backbone, Îâzfbulk=FΞâ(zft) z_f^bulk=F_Ξ(z_f^t), which models domain-wide liquid evolution. U-shaped neural operators and U-FNO-style models have been used to improve multiscale field prediction and memory efficiency in neural operator learning (Rahman et al., 2022; Wen et al., 2022). The interface update is produced by encoding ât+1I_t+1 into an interface latent, updating the recurrent interface state, and injecting the resulting interface information back into the fluid grid through a localized mediator. This gives Îâzfint z_f^int, the latent residual induced by the moving solid. The tentative latent is then stabilized before transport: zÂŻft+1=Î lat,1â(z~ft+1), z_f^t+1= _lat,1( z_f^t+1), (3) where Î lat,1 _lat,1 is a learned residual correction operator. From this corrected latent, TIDES decodes a provisional velocity field and uses it to advect auxiliary particles. These particles are not the prediction target; they act as Lagrangian probes of the transport implied by the Eulerian latent state. Similar Lagrangian or particle-based representations have been useful for learning local transport and liquidâobject interactions (Pfaff et al., 2020; Midlagajni and Rothkopf, 2025). The resulting particle displacement is rasterized back to the voxel grid and fused into the fluid latent: zft+1=Î lat,2â(zÂŻft+1+RP2Gâ(pt,pt+1,ât+1)).z_f^t+1= _lat,2 ( z_f^t+1+R_P2G(p_t,p_t+1,I_t+1) ). (4) Here, RP2GR_P2G converts particle displacement and local particle motion statistics into a grid-aligned transport feedback signal, and Î lat,2 _lat,2 is a second learned correction operator applied after this feedback. This post-transport correction helps reduce error accumulation during long-horizon rollout, which is especially important in autoregressive scientific forecasting (Koehler et al., 2024). Finally, TIDES decodes the stabilized latent and applies an interface-aware output correction: u^t+1=Î outâ(Dvâ(zft+1),ât+1). u_t+1= _out (D_v(z_f^t+1),I_t+1 ). (5) The output correction is a convolutional velocity correction conditioned on the same interface representation. Thus, TIDES does not replace the localized principle of TIDE; it strengthens it with latent correction, particle-grid transport feedback, and output-level stabilization. 3.4 Autoregressive Rollout and Objective The model is autoregressive only in the liquid state; the future solid sequence is supplied at every step: u^t+k+1=fΞ(u^t+k,Ït+k+1s,vt+k+1s,pt+k),k=0,âŠ,Tâ1,u^t=ut. u_t+k+1=f_Ξ( u_t+k, _t+k+1^s,v_t+k+1^s,p_t+k), k=0,âŠ,T-1, u_t=u_t. (6) Thus, the liquid state is recursively updated from the modelâs own predictions, while the solid trajectory remains an external conditioning signal. This follows the broader autoregressive emulator setting used in learned PDE forecasting benchmarks (Koehler et al., 2024). When the dual-space branch is enabled, the particle state is updated jointly with the Eulerian rollout; otherwise, the target remains only the dense Eulerian velocity sequence. We train with five functional loss groups: â=âfield+λrollââroll+λintââint+λtransââtrans+λstabââstab.L=L_field+ _rollL_roll+ _intL_int+ _transL_trans+ _stabL_stab. (7) âfieldL_field anchors one-step dense velocity prediction, ârollL_roll supervises multi-step autoregressive quality, âintL_int emphasizes interface-local fidelity, âtransL_trans aligns Eulerian prediction with Lagrangian transport behavior, and âstabL_stab regularizes stable rollout. Rollout-based objectives and long-horizon metrics are important for evaluating temporal generalization in neural PDE emulators. In the full implementation, these groups decompose into several auxiliary losses, including dense field losses, rollout losses, interface-weighted losses, particle/transport consistency terms, and correction regularizers. Appendix E.5 gives the exact decomposition and weights. 4 Experiments 4.1 Task Setup and Evaluation We evaluate moving-solid-conditioned interactive 4D volumetric liquid forecasting: given an initial liquid state and a solid condition, the model predicts a dense 3D Eulerian liquid-velocity rollout on a 32Ă128Ă12832Ă 128Ă 128 grid at 8 FPS. The benchmark contains sphere, cube, and cone object families with motion factors spanning path length, speed, acceleration/deceleration, and straight or curved trajectories. We use 12800 / 1600 / 1600 / 4000 episodes for train / validation / test / tâeâsâtoodtest_ood. All core comparisons use the same rollout-oriented training setup, optimization budget, and evaluation protocol; the main comparison reports mean performance over 5 random seeds with 10000 update steps per seed. Models are trained and tested at rollout horizon 8, with horizon extension evaluated at rollout lengths 16 and 24. Motion OOD uses unseen combinations of motion and trajectory factors, while size OOD doubles sphere radius, cone base radius, and cube edge length at test time. The dense target is a benchmark-defined Eulerian velocity field constructed from particle simulations by trilinear/CIC particle-to-grid conversion, not a solver-native continuum field. Exact metric definitions, fallback mask rules, and particle-to-grid details are given in Appendix D. Because the task is motivated by repeated querying, Appendix C also reports latency: TIDES takes 302.9 ms for an 8-step rollout on the benchmark hardware, compared with about 6.46 s of DualSPHysics simulation time per physical second for representative benchmark cases. We use this as a forecasting-speed reference only, not as a claim of simulator equivalence. Development-time variants such as interface-near-aware and t+1t+1-focused objectives are excluded from the primary evidence because they change the final method definition. Because no standard benchmark exists for this exact task, we use a metric suite adapted from autoregressive PDE forecasting, transport-sensitive evaluation, and physics-aware diagnostics. We treat rollout field error (lrolll_roll), near-interface field error (nearnear), and transport endpoint error (târâaâjendtraj_end) as the primary forecasting metrics, since they measure overall rollout quality, interaction-region accuracy, and material transport consistency. We additionally report far-field error (farfar) for propagation behavior and divergence delta (Îdiv _div) as a stabilization-oriented diagnostic, not as a standalone measure of physical fidelity. Since field metrics are normalized voxel averages, we emphasize consistent ranking, cross-metric tradeoffs, and robustness under OOD and long-horizon evaluation rather than raw decimal scale alone. 4.2 Interface-Localized Response Characterization We first characterize how moving-solid interaction affects volumetric liquid response. Here, response energy denotes the per-voxel squared temporal velocity change, âutâ(x)âutâ1â(x)â22\|u_t(x)-u_t-1(x)\|_2^2, averaged over valid liquid voxels within each relative distance bin from the moving-solid interface. Table 1 shows that this response energy is concentrated near the evolving interaction interface and drops by about 455Ă455Ă from the nearest interface band to the far field. This motivates treating the task as interface-driven forecasting rather than bulk-only prediction. Table 1: Distance-to-interface response energy. Lower interface distance corresponds to voxels closer to the evolving interaction boundary. Relative interface distance bin Mean response energy [0.00,0.05)[0.00,0.05) 1.663763 [0.05,0.10)[0.05,0.10) 0.344675 [0.10,0.20)[0.10,0.20) 0.104469 [0.20,0.35)[0.20,0.35) 0.012967 [0.35,0.50)[0.35,0.50) 0.004842 [0.50,1.00)[0.50,1.00) 0.003659 4.3 Main Comparison Since no standard benchmark or exact task-matched baseline exists, Table 2 is organized to answer complementary comparison questions rather than to present a single exact-competitor ranking. Bulk Only tests whether dense liquid rollout can be predicted without explicit moving-solid conditioning. The adapted 3D ResUNet, U-NO, Transolver, and GNS baselines anchor the benchmark against neighboring volumetric, neural-operator, transformer-based, and particle model families (Rahman et al., 2022; Wu et al., 2024; Pfaff et al., 2020; Midlagajni and Rothkopf, 2025). Our variants then test the modeling hierarchy within the proposed family: TIDE uses the localized interface-driven core, Finite Support widens the localized residual support as an exploratory variant, and TIDES adds transport-aware feedback and stabilization. All voxel models are trained under the same dense target, rollout horizon, optimization budget, and metric protocol, with 3D ResUNet receiving a small proxy width sweep before full-budget retraining. GNS preserves native particle-state prediction and is evaluated by rasterizing predicted particle velocities to the common voxel grid, following the broader use of particle/graph simulators for learned dynamics (Pfaff et al., 2020; Midlagajni and Rothkopf, 2025). We deliberately do not attach our localized mediator, transport feedback, or correction modules to the external baselines, since doing so would turn them into backbone-swapped versions of our method rather than neighboring references. Further adaptation details are given in Appendix A and Table 6. Table 2: Main comparison under the same optimization budget and evaluation protocol, aggregated over 5 random seeds. Bold indicates the best performance. Model lrollâl_roll târâaâjendâtraj_end nearânear farâfar Îdiv _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 TIDE 0.006113 0.110904 0.005660 0.006279 -0.000248 TIDES 0.005746 0.104397 0.005581 0.005826 -0.057067 Figure 3: Qualitative comparison of autoregressive velocity rollouts. Each column shows a model and each row shows a rollout step from t to t+16t+16. The white contour indicates the moving-solid location,and brighter colors indicating higher velocity magnitude. The main pattern is the progression from no explicit interaction modeling to localized interaction modeling and then to stabilized transport-aware rollout. Bulk Only is weaker than the interaction-aware variants, showing that the moving solid cannot be treated as incidental context. TIDE improves the primary forecasting metrics relative to the adapted external references and the widened-support variant, which supports localized interface conditioning as the core inductive bias. TIDES further reduces rollout, interface, far-field, and trajectory errors while producing the clearest stabilization diagnostic, suggesting that transport-aware feedback and correction modules matter once the localized core is used autoregressively. 4.4 Localized Principle Validation We next isolate the localized-interaction design. Table 3 compares TIDE with variants that remove projection, remove object-local context, replace localized residual updates with global conditioning, or widen the interface support. Table 3: Localized-centered principle ablation. Bold indicates the best performance. Variant lrollâl_roll târâaâjendâtraj_end nearânear farâfar Îdiv _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 TIDE gives the best primary metrics in this group. Removing projection produces the largest degradation, indicating that post-update consistency matters even for the localized core. Removing object-local context, replacing localized residuals with global conditioning, or widening the support also degrades performance, supporting the default narrow-band localized design. 4.5 Full-Model Tradeoffs Table 4: Full-model component ablation. Bold indicates the best performance. Variant lrollâl_roll târâaâjendâtraj_end nearânear farâfar Îdiv _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 We next remove one full-model component at a time from TIDES: dual transport feedback, latent correction, output correction, or both correction stages together. Show in Table 4, TIDES is best across the primary metrics, and each removal weakens performance. The largest drop comes from removing dual feedback, which sharply worsens transport endpoint accuracy. Removing latent or output correction also degrades the metrics, while removing both corrections causes a larger transport failure. This pattern supports the view that transport feedback and correction stages are complementary rather than a single interchangeable addition. 4.6 Objective Complementarity 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. Variant lrollâl_roll târâaâjendâtraj_end nearânear farâfar Îdiv _div Full Loss 0.005746 0.104397 0.005581 0.005826 -0.057067 w/o LtrajL_traj 0.005801 0.110317 0.005768 0.005951 -0.030720 w/o LintL_int 0.005867 0.105963 0.005638 0.005903 -0.045270 w/o LrollL_roll 0.006907 0.105727 0.006012 0.007216 -0.041455 Table 5 fixes the TIDES architecture and removes one loss term at a time. The complete objective performs best across the primary metrics. Removing LtrajL_traj weakens transport accuracy, removing LintL_int degrades interaction-sensitive error and stabilization, and removing LrollL_roll causes the clearest overall deterioration, especially on lrolll_roll and farfar. Exact implementation-level loss decompositions and weights are listed in Appendix E.5. 4.7 OOD and Long-Horizon Forecasting We evaluate two distribution shifts and one horizon extension. Motion/trajectory OOD uses unseen combinations of path length, speed, acceleration/deceleration, and straight/curved trajectories. Size OOD doubles sphere and cone radius and cube edge length at test time while keeping all models trained on the original dense benchmark. Horizon extension evaluates models trained at rollout length 8 at rollout lengths 16 and 24. Tables 10, 11, and 12 in appendix G report these results. The OOD and horizon results are consistent with the main comparison. TIDES gives the best aggregate behavior on motion OOD, size OOD, and both longer horizons, with the clearest stabilization diagnostic. TIDE remains competitive as the localized core, while external baselines can match isolated metrics but do not match the combined rollout, transport, and stabilization behavior. These results suggest that transport-aware coupling and explicit stabilization become most useful as distribution shift or accumulated rollout error increases. 5 Discussion Our results support a local-to-global view of moving-solid-conditioned liquid forecasting: the interaction is initiated at the evolving solidâliquid interface, but the target is a full-domain Eulerian rollout. This is consistent with classical and learned FSI perspectives, where solidâfluid exchange is mediated through moving interfaces or boundary conditions (Kolahdouz et al., 2021; Zhong and Meidani, 2025; Tao et al., 2026). The experiments indicate that localized interface modeling provides the core forecasting mechanism, while transport-aware feedback and explicit stabilization improve robustness under autoregressive rollout, OOD motion/size shifts, and horizon extension (Koehler et al., 2024; Yagoubi et al., 2025; Hu et al., 2026). 6 Limitations This benchmark should be interpreted as a controlled study rather than a replacement for physical simulation. The data are synthetic, the OOD axes are limited to motion and object-size shifts, and the dense Eulerian target is constructed from particle simulations rather than measured as a solver-native continuum field. Similar sim-to-real and benchmark-coverage limitations have been discussed in recent scientific ML benchmarks (Hu et al., 2026; Yang et al., 2025). Future work should extend the task to richer geometries, broader physical parameter shifts, multi-solid interactions, and more practical repeated-query liquid forecasting. 7 Conclusion We formulated interactive 4D volumetric liquid forecasting as a moving-solid-conditioned dense prediction task: given a current liquid state and a known future solid motion, the model forecasts the future full-domain Eulerian liquid velocity rollout. Across the controlled benchmark, our results identify localized interface-driven interaction as the central modeling principle, and show that transport-aware coupling and explicit stabilization improve long-horizon autoregressive forecasting. We hope this task, benchmark, and TIDE/TIDES study provide a foundation for broader research on motion-conditioned liquid forecasting under richer geometries, broader distribution shifts, and more practical repeated-query settings. Acknowledgments and Disclosure of Funding This work was partly supported by the Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) [RS-2021-I211341, Artificial Intelligence Graduate School Program (ChungAng University)] and partly supported by the âAdvanced GPU Utilization Support Programâ funded by the Government of the Republic of Korea (Ministry of Science and ICT). References Akkerman et al. (2025) R. Akkerman, H. Feng, M. J. Black, D. Tzionas, and V. F. Abrevaya Interdyn: controllable interactive dynamics with video diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, p. 12467â12479. Cited by: §1, §1. Du et al. (2024) P. Du, M. H. Parikh, X. Fan, X. Liu, and J. Wang Conditional neural field latent diffusion model for generating spatiotemporal turbulence. Nature Communications 15 (1), p. 10416. Cited by: §1. Ferdian et al. (2020) E. Ferdian, A. 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Xie Multiphysics bench: benchmarking and investigating scientific machine learning for multiphysics pdes. arXiv preprint arXiv:2505.17575. Cited by: §1, §6. Zhong and Meidani (2025) W. Zhong and H. Meidani Benchmarking of neural operators for rigid-body fluid-structure interaction. Harvard Dataverse dataset, p. 1141. Cited by: §1, §1, §2, §3.1, §3.2, §5. Appendix A Adaptation Details for External Baselines All external baselines are trained against the same dense voxelized forecasting target used in the main benchmark: next-step Eulerian liquid velocity on the shared grid. The two voxel-space baselines, 3D ResUNet and U-NO, receive the current liquid velocity together with global grid-aligned solid-conditioning channels derived from the future solid state. These channels encode solid-state cues such as boundary velocity and interface proximity, but the models do not use the localized mediator pathway, transport-feedback pathway, or explicit stabilization modules of the finalized method. For Transolver, we use the same voxelized liquid input but concatenate a global solid-conditioning grid composed of voxelized boundary velocity and fluidâsolid interface channels before applying full-grid physics-attention. For GNS, we preserve native Lagrangian particle prediction by representing the moving solid as kinematic rigid proxy particles and compute voxel-space metrics only after rasterizing predicted particle velocities onto the shared Eulerian grid. Unless otherwise noted, the same dense target-construction protocol is used for all methods so that every baseline is scored in the same prediction space. Table 6 summarizes the input, conditioning, output, and evaluation space used for each adapted external baseline. Table 6: Implementation details for adapted external baselines under the dense-liquid, simplified-input protocol. 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 B Simulator Physics Behind Dataset Generation The reported dense-liquid benchmark is generated from DualSPHysics 5.4, a weakly compressible smoothed particle hydrodynamics (WCSPH) NavierâStokes solver for free-surface flow and fluid-driven rigid bodies. In our pipeline, case geometry is discretized by GenCase, the fluid simulation itself is executed by DualSPHysics5.4, fluid particles are exported by PartVTK, and rigid-body motion is exported by FloatingInfo. The resulting particle trajectories are then converted into the shared dense Eulerian benchmark target described in Appendix D. At the continuum level, the solver advances a Lagrangian free-surface flow model governed by mass and momentum conservation, dâÏdât=âÏââ u,dâudât=â1Ïâp+g+Î, dÏdt=-Ïâ· u, dudt=- 1Ïâ p+g+ , where Ï is density, u velocity, p pressure, g gravity, and Î denotes dissipative terms such as viscosity. DualSPHysics discretizes these equations with SPH particle summations over a compact-support kernel. Pressure is obtained from a weakly compressible equation of state of Tait type, p=BâĄ[(Ï0)Îłâ1],p=B [ ( Ï _0 )^Îł-1 ], so that the simulation avoids a global pressure Poisson solve while keeping density fluctuations small when the artificial sound speed is sufficiently high. In the reported rigid-object cases, the case XML sets Ï0=1000âkg/m3 _0=1000\,kg/m^3, Îł=7Îł=7, gravity g=(0,0,â9.81)g=(0,0,-9.81), coefsound=20, hdp=1.3, and cflnumber=0.2. The reported runs use the following DualSPHysics numerical ingredients. The smoothing kernel is the Quintic Wendland kernel. Dissipation is handled with Monaghan artificial viscosity. Density noise is reduced with the Fourtakas density-diffusion term. Particle disorder is controlled with the Lind et al. shifting algorithm. Time integration uses the symplectic position-Verlet scheme with a CFL-controlled variable timestep. Rigid-body motion is treated through the built-in floating-body formulation, in which pressure and viscous forces on boundary particles are accumulated into net force and torque and then integrated with rigid-body dynamics. In our reported rigid-object benchmark, the object is therefore not replayed as a purely kinematic animation; it remains fluid-coupled through the floating-body solver, while its nominal forcing schedule is generated by the runtime case-preparation script. One implementation detail matters for interpreting boundary accuracy. The template XML is authored with Boundary=2, i.e. the modified dynamic boundary condition (mDBC), but the runtime case script switches the reported rigid closed-tank runs to Boundary=1, i.e. the classic dynamic boundary condition (DBC), because mDBC normal generation can fail in this enclosed floating-body setup. Thus, the benchmark should be understood as using the standard DualSPHysics WCSPH free-surface formulation together with a robust DBC rigid-wall treatment for the finalized rigid-object dataset. This is still a physically coupled CFD simulation, but it is not the most aggressive boundary-accuracy setting available in the general solver. The benchmark input contract should be distinguished from the simulator internals. During simulation, the rigid object follows a realized fluid-coupled rigid-body trajectory under the runtime forcing schedule. After simulation, this realized future solid-state sequence is recorded and provided to the forecasting model as known conditioning. The model therefore receives future solid states but never future liquid states. In this sense, the benchmark assumes a known future moving-solid sequence and evaluates only the problem of forecasting the liquid response to that sequence. We therefore treat the simulator output as a high-fidelity reference rather than as exact ground truth. This is justified because DualSPHysics is a mature research and engineering code with extensive validations against analytical solutions, numerical benchmarks, and experimental data, and because WCSPH is particularly well matched to violent free-surface flow, splashing, and moving solid interaction. At the same time, the reference remains a numerical approximation: it inherits the assumptions of weak compressibility, artificial dissipation, boundary modelling, particle shifting, and finite resolution. In addition, our benchmark applies a separate particle-to-voxel target-construction stage, and some dataset-generation variants optionally add low-amplitude ambient surface perturbations during post-processing. For transparency, the dataset contract stores both raw and post-processed velocity fields when such perturbations are used, so that solver-native and augmented signals remain distinguishable. Appendix C Efficiency and Reference-Simulator Latency Table 7 reports the rollout-8 inference latency of the learned models together with a reference DualSPHysics timing derived from two representative benchmark logs. We use the simulatorâs reported simulation runtime per physical second as the reference quantity and compare it against model inference for an 8-step rollout, since the finalized benchmark is sampled at 8 FPS. This comparison should be interpreted only as a controlled repeated-query latency reference under the benchmark contract, not as a claim that learned forecasting is equivalent to full CFD simulation. 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. Method Rollout-8 latency Relative speed vs. DualSPHysics DualSPHysics (reference simulator) 6.46 s 1.0Ă 3D ResUNet 130.2 ms 49.6Ă U-NO 201.9 ms 32.0Ă GNS 77.4 ms 83.5Ă Transolver 313.5 ms 20.6Ă TIDE 202.0 ms 32.0Ă TIDES 302.9 ms 21.3Ă The table makes two points relevant to the paperâs positioning. First, TIDES remains substantially faster per query than the reference simulator under the benchmark protocol. Second, the more structured transport-aware models incur a meaningful latency cost relative to simpler voxel baselines, so the benchmark should be read as exposing an accuracyâlatency tradeoff rather than as identifying a universally dominant surrogate. Appendix D Exact Metric Definitions and Particle-to-Grid Conversion This section records the exact metric definitions and dense target-construction rules used in the reported experiments. Rollout field error. Let u^t u_t and utu_t denote the predicted and ground-truth Eulerian velocity fields at rollout step t. We first compute a per-step field error et=meanc,d,h,wâââ(u^tâut),without spatial weighting,âââĄ(u^tâut)âwspatialâwspatial,with spatial weighting,e_t= casesmean_c,d,h,w\, ( u_t-u_t),&without spatial weighting,\\[4.0pt] ÎŁ ( u_t-u_t)\,w_spatialÎŁ w_spatial,&with spatial weighting, cases where ââĄ(x)=|x| (x)=|x| or x2x^2 depending on the training configuration. The rollout metric is then lroll=1Bââb=1Bât=1TÎłtâ1âet(b)ât=1TÎłtâ1.l_roll= 1B _b=1^B _t=1^TÎł^t-1e_t^(b) _t=1^TÎł^t-1. In the current setting, Îł=1Îł=1, so lrolll_roll reduces to the step-wise average rollout field error. Near/Far velocity error. Let S=|Ïs|S=|Ï^s| denote the absolute solid-distance field and let M denote the valid dense liquid-domain mask. In the dense-liquid setting, M is the full non-solid liquid domain rather than the sparse particle-support mask. The default near/far masks are defined by Mnear=[Sâ€Ïnear]â M,Mfar=[Sâ„Ïfar]â M,M_near=1[S†_near]· M, M_far=1[Sâ„ _far]· M, with default thresholds Ïnear=0.2 _near=0.2 and Ïfar=0.9 _far=0.9. If these masks become degenerate or overlap excessively, we fall back to quantile-based thresholds Ïn=Q0.25â(SâĄ[M]),Ïf=Q0.75â(SâĄ[M]), _n=Q_0.25(S[M]), _f=Q_0.75(S[M]), and define masks accordingly. If this still fails, we apply a rank-based split over valid voxels, assigning the lowest 25% of interface distances to near and the highest 25% to far. The corresponding metrics are masked velocity errors: near=ErrâĄ(u^,u,Mnear),far=ErrâĄ(u^,u,Mfar).near=Err( u,u;M_near), =Err( u,u;M_far). Trajectory endpoint error. Let p^T p_T be particles advected by the predicted velocity field and pTp_T the ground-truth particle endpoints. We define târâaâjend=1Nââi=1Nâp^T,iâpT,iâ2.traj_end= 1N _i=1^N\| p_T,i-p_T,i\|_2. Divergence delta. Let ustage1â_âpreu_stage1\_pre denote the velocity before the final stabilization / projection stage and let ufinalu_final denote the final projected output. With DivScoreâĄ(u)=meanâĄ(|ââ u|),DivScore(u)=mean (|â· u| ), we define Îdiv=DivScoreâĄ(ufinal)âDivScoreâĄ(ustage1â_âpre). _div=DivScore(u_final)-DivScore(u_stage1\_pre). Thus, more negative values indicate larger divergence reduction and therefore better stabilization. Dense target construction from particles. Dataset construction begins with fluid-particle trajectories from the simulator. Let pip_i and viv_i denote particle positions and velocities. We first voxelize particles using trilinear/CIC splatting onto the Eulerian grid: Wg=âiwiâg,Vg=âiwiâgâvi,W_g= _iw_ig, V_g= _iw_igv_i, and define a sparse particle-supported velocity field by ugsparse=Vg/Wg,Wg>Ï”,0,otherwise.u_g^sparse= casesV_g/W_g,&W_g>Δ,\\ 0,&otherwise. cases This yields a sparse support mask Msupp=[Wg>Ï”].M_supp=1[W_g>Δ]. In the dense-liquid setting, the final training target is not this sparse support field. Instead, we define the valid dense liquid domain as the full non-solid region Mliq=1âMsolid,M_liq=1-M_solid, optionally clipped by a fixed water-level ceiling when constructing the benchmark. We then extend the sparse field over the valid dense liquid domain using nearest-support interpolation: ugdense=uÏâĄ(g)sparse,ÏâĄ(g)=argâĄmingâČâMsuppâĄâgâgâČâ2,gâMliq.u_g^dense=u_Ï(g)^sparse, Ï(g)= _g â M_supp\|g-g \|_2, gâ M_liq. Outside the valid liquid domain, the target velocity is set to zero. Thus, the dataset stores both the sparse particle-support mask and the final dense target field, but the reported models are trained on the dense full-domain Eulerian target. This field should be read as a benchmark-defined dense Eulerian forecasting target constructed from particle simulations, not as an exact solver-native continuum field. Different P2G operators used in the system. The dataset voxelizer above should not be confused with the internal dual-space transport feedback operator or the GNS evaluation voxelization. The dataset target constructor uses CIC splatting followed by dense-domain filling; the internal dual-space P2G uses nearest-voxel scatter of particle displacements; and the GNS evaluation pipeline uses a separate voxelization routine for converting predicted particle states into the shared Eulerian evaluation space. Appendix E Implementation Details of the Reported Dense-Simplified Configuration E.1 Operator Types and Tensor Shapes The finalized dense-simplified configuration takes the current liquid velocity field together with a known future solid-state sequence. In the reported setting, the dense fluid tensor is xtââBĂ3ĂDĂHĂW,(D,H,W)=(32,128,128),x_t ^BĂ 3Ă DĂ HĂ W, (D,H,W)=(32,128,128), containing only the current Eulerian liquid velocity components. The solid sequence is St+1:t+T=(Ït+1s,vt+1s),âŠ,(Ït+Ts,vt+Ts),S_t+1:t+T=\( _t+1^s,v_t+1^s),âŠ,( _t+T^s,v_t+T^s)\, where ÏÏsââBĂ1ĂDĂHĂW,vÏsââBĂ3ĂDĂHĂW. _Ï^s ^BĂ 1Ă DĂ HĂ W, v_Ï^s ^BĂ 3Ă DĂ HĂ W. From each solid state, the implementation derives a 10-channel interface-aligned tensor consisting of fluid-solid interface, liquid-liquid interface, interface sdf, interface normal, boundary velocity, and boundary mask, where the liquidâliquid interface channel is zero in the dense-liquid setting and is retained only for channel compatibility. Legacy object-centric descriptors and control tensors remain available in the codebase for backward compatibility and exploratory ablations, but they are not part of the reported simplified input API. The auxiliary particle state used in the Full Stabilized model is ptââBĂPĂ3,P=256.p_t ^BĂ PĂ 3, P=256. When the explicit dual-space branch is enabled, the predicted particle trajectory tensor has shape p^1:TââBĂ(T+1)ĂPĂ3. p_1:T ^BĂ(T+1)Ă PĂ 3. The fluid backbone is implemented as a 3-stage UNOOperator3D. Each stage uses a spectral-local residual block consisting of a 3D spectral convolution, a pointwise 1Ă1Ă11Ă 1Ă 1 convolution, a local residual 3D convolution block, group normalization, and GELU activation. The interface branch is implemented by InterfaceBranchEncoder3D. In the reported setting, it receives 10 interface channels, applies a 3D convolutional stem, and tokenizes the result through adaptive pooling and linear token projection. The interface token grid has spatial resolution (2,4,4)(2,4,4), yielding 32 global interface tokens in the default configuration. In the reported TIDES configuration, the simplified input pathway does not require a separate object-patch token. Object-local patch context appears only in the TIDE localized-family variants used in the localized-principle ablation. The rollout core is implemented so that solid influence is mediated through the derived interface pathway rather than through naive global injection. E.2 Interaction Operator and One-Step Coupling Structure The finalized full model uses a structured interface-mediated interaction operator rather than naive global conditioning. At each step, the next-step solid state is first converted into a derived interface representation, which is then encoded into interface tokens. Let zinttz_int^t denote the interface tokens and zftz_f^t the fluid latent grid. The implemented update can be summarized as ât+1=âĄ(Ït+1s,vt+1s),z~intt+1=Eintâ(ât+1),I_t+1=D( _t+1^s,v_t+1^s), z_int^t+1=E_int(I_t+1), zintt+1=zintt+Attnintâfâ(zft)+z~intt+1.z_int^t+1=z_int^t+Attn_intâ f(z_f^t)+ z_int^t+1. The mediator pathway then performs interface-to-fluid exchange: Îâzintâft=Attnfâintâ(zintt+1), z_intâ f^t=Attn_f (z_int^t+1), which is reconstructed back to a grid-aligned latent residual using a token-to-grid path: Îâzintt=ImpulseRefineâĄ(TokensToGridâĄ(Îâzintâft)). z_int^t=ImpulseRefine\! (TokensToGrid( z_intâ f^t) ). A learned interaction-intensity head produces a multiplicative amplification factor, It=hintensityâ(|Îâzintt|),ÎâzinttâÎâzinttâ(1+It).I_t=h_intensity\! (| z_int^t| ), z_int^tâ z_int^t (1+I_t). Bulk latent transport is computed independently by the fluid U-NO backbone: Îâzbulkt=FΞâ(zft). z_bulk^t=F_Ξ(z_f^t). The combined latent update before correction is therefore z~ft+1=zft+Îâzbulkt+ÎșâÎâzintt. z_f^t+1=z_f^t+ z_bulk^t+Îș z_int^t. Thus, in the reported TIDES model, solid influence is not injected as an independent global control signal; instead, future solid geometry and motion are routed through the derived interface representation and only then reconstructed as a localized grid-aligned latent impulse. E.3 Transport Coupling and Correction Modules The dual-space coupling pathway uses a grid-to-particle (G2P) update followed by a particle-to-grid (P2G) feedback step. First, a velocity-like field is decoded from the corrected latent state and sampled at particle locations using trilinear interpolation: uliket=Dvâ(zf,lat1t+1),pt+1(n)=clipâĄ(pt(n)+âĄ(uliket,pt(n))),u_like^t=D_v(z_f,lat1^t+1), p_t+1^(n)=clip\! (p_t^(n)+S(u_like^t,p_t^(n)) ), where S denotes trilinear sampling. The updated particle positions are then rasterized back to the grid through nearest-voxel scatter. Let gn=roundâĄ(pt+1(n)).g_n=round(p_t+1^(n)). The model accumulates a density grid and displacement-sum grid: Ï(g)=ân[gn=g],dsum(g)=ân(pt+1(n)âpt(n))[gn=g].Ï(g)= _n1[g_n=g], d_sum(g)= _n(p_t+1^(n)-p_t^(n))1[g_n=g]. The mean displacement grid is dÂŻâ(g)=dsumâ(g)maxâĄ(ÏâĄ(g),1), d(g)= d_sum(g) (Ï(g),1), and the density grid is normalized by the maximum density within each sample: Ï~â(g)=ÏâĄ(g)maxgâČâĄÏâĄ(gâČ). Ï(g)= Ï(g) _g Ï(g ). The dual feedback tensor is then constructed by concatenating normalized density, mean displacement, and interface context: zp2gt=DualRefineâĄ([Ï~,dÂŻ,IntCtx]),z_p2g^t=DualRefine\! ([ Ï, d,IntCtx] ), where DualRefine is a learned 3D convolutional refinement block. This correction is added back to the fluid latent grid before the second latent correction stage. E.4 Latent and Output Stabilization Modules The latent correction modules Î lat,1 _lat,1 and Î lat,2 _lat,2 are both implemented by LatentPressureCorrectionOperator. These are not hard Poisson solvers; rather, they are lightweight learned residual correction modules with pressure-like structure. Given an input latent tensor z, the operator first injects a global pooled summary: zglob=z+Wgâ(AvgPool3DâĄ(z)),z_glob=z+W_g\! (AvgPool3D(z) ), where the pooled summary is broadcast back to the original spatial resolution through a learned 1Ă1Ă11Ă 1Ă 1 transform. A scalar correction potential is then predicted: Ï=WÏ,2â(GELUâĄ(WÏ,1â(zglob))),ÏââBĂ1ĂDĂHĂW.Ï=W_Ï,2\! (GELU\! (W_Ï,1(z_glob) ) ), Ï ^BĂ 1Ă DĂ HĂ W. Its finite-difference gradient is computed as âÏ=[ÎxâÏ,ÎyâÏ,ÎzâÏ],âÏ=[ _xÏ, _yÏ, _zÏ], with forward finite differences and zero padding at the spatial boundary. The correction is mapped back to latent space by a learned 1Ă1Ă11Ă 1Ă 1 transform: corrâĄ(z)=Wââ(âÏ),corr(z)=W_â(âÏ), and the corrected latent tensor is Î latâ(z)=zglobâαâcorrâ(z). _lat(z)=z_glob-α\,corr(z). The first latent correction stage is applied before dual-space P2G feedback, zf,lat1t+1=Î lat,1â(z~ft+1),z_f,lat1^t+1= _lat,1( z_f^t+1), and the second is applied after adding dual feedback, zf,lat2t+1=Î lat,2â(zf,lat1t+1+ÎČp2gâzp2gt).z_f,lat2^t+1= _lat,2\! (z_f,lat1^t+1+ _p2gz_p2g^t ). The output correction module receives the tentative decoded velocity field together with derived interface and boundary cues: xout=[u~t+1,sint,mbnd,ubnd,It],x_out=[ u_t+1,s_int,m_bnd,u_bnd,I_t], for a total of 9 input channels. The correction network consists of Conv3DâGNâGELUâConv3DâGNâGELUâConv1Ă1Ă1,Conv3D 3D 1Ă 1Ă 1, producing a velocity correction Îâu u. The corrected output is ut+1corr=u~t+1+ÎČoutâÎâu,u_t+1^corr= u_t+1+ _out u, followed by hard boundary overwrite: u^t+1=(1âmbnd)âut+1corr+mbndâubnd. u_t+1=(1-m_bnd) u_t+1^corr+m_bnd u_bnd. A small feedback term derived from u^t+1âu~t+1 u_t+1- u_t+1 and the interaction-intensity field is also fed back to the latent state. E.5 Loss Decomposition and Weights While the main text groups the training objective into functional categories, the dense-simplified implementation uses the loss bundle âimpl= _impl= λvelââvel+λgradââgrad+λdivââdiv+λtrajââtraj+λintââint _velL_vel+ _gradL_grad+ _divL_div+ _trajL_traj+ _intL_int +λrollââroll+λphaseââphase+λdualââdual+λexchangeââexchange+λparticleââparticle. + _rollL_roll+ _phaseL_phase+ _dualL_dual+ _exchangeL_exchange+ _particleL_particle. In the default dense-simplified configuration, λvel=1.0,λgrad=0.2,λdiv=0.1,λtraj=0.5,λint=0.2, _vel=1.0, _grad=0.2, _div=0.1, _traj=0.5, _int=0.2, λroll=0.5,λphase=0.1,λdual=0.5,λexchange=0.1,λparticle=0.2. _roll=0.5, _phase=0.1, _dual=0.5, _exchange=0.1, _particle=0.2. Table 8 summarizes the implementation-level loss weights used by the dense-simplified full model. Table 8: Loss terms used in the dense-simplified implementation. Loss term Symbol Weight Velocity reconstruction λvel _vel 1.0 Gradient / vorticity consistency λgrad _grad 0.2 Divergence regularization λdiv _div 0.1 Trajectory supervision λtraj _traj 0.5 Interface consistency λint _int 0.2 Rollout consistency λroll _roll 0.5 Phase supervision λphase _phase 0.1 Dual-space consistency λdual _dual 0.5 Mediator exchange balance λexchange _exchange 0.1 Explicit particle-branch supervision λparticle _particle 0.2 Here, âvelL_vel is the one-step dense Eulerian reconstruction loss, âgradL_grad matches spatial velocity derivatives and optional vorticity, âdivL_div is the mean absolute divergence regularizer on the predicted rollout, âtrajL_traj supervises particles obtained by advecting sampled particles through the predicted Eulerian field, and ârollL_roll is the multi-step weighted dense rollout error. In the reported benchmark, the phase term should be read narrowly: it is an auxiliary two-channel liquid/solid occupancy supervision used in the full-model family, rather than a general multi-phase prediction objective used by every baseline. Simpler localized and external baselines disable this head. The interface loss is âint=wbvââboundaryâ_âvelocity+wnpâânoâ_âpenetration+wnâânormalâ_âconsistency.L_int=w_bvL_boundary\_velocity+w_npL_no\_penetration+w_nL_normal\_consistency. The dual-space consistency term âdualL_dual measures EulerianâLagrangian consistency under the same sampled particles, âexchangeL_exchange penalizes imbalance in the mediator exchange pathway, and âparticleL_particle supervises the explicit particle rollout branch when present. For presentation in the main paper, these implementation-level terms are grouped into the five functional categories of field reconstruction, rollout consistency, interface consistency, transport consistency, and stabilization. Simpler baselines and ablations disable the terms that are irrelevant to their architecture. Appendix F Seed Stability of the Main Comparison Table 9 reports the same main-comparison means as Table 2, together with standard deviations across 5 random seeds. We include this table to make the seed variability explicit while keeping the main text focused on the aggregate comparison. Table 9: Seed-stability version of Table 2. The mean values match the main-text table, and the appendix additionally reports mean ± standard deviation across 5 random seeds. Lower is better. Model lrollâl_roll trajâ_âendâtraj\_end nearânear farâfar Îdiv _div 3D ResUNet 0.0075850.007585 ±0.0011± 0.0011 0.1227810.122781 ±0.1412± 0.1412 0.0057570.005757 ±0.0015± 0.0015 0.0095530.009553 ±0.0031± 0.0031 +0.000000+0.000000 ±0.0007± 0.0007 U-NO 0.0065420.006542 ±0.0005± 0.0005 0.1121110.112111 ±0.1679± 0.1679 0.0058070.005807 ±0.0020± 0.0020 0.0059990.005999 ±0.0021± 0.0021 â0.000376-0.000376 ±0.0008± 0.0008 GNS 3.7671703.767170 ±0.0286± 0.0286 5.5552105.555210 ±1.8474± 1.8474 3.6735783.673578 ±0.0286± 0.0286 3.9575123.957512 ±0.0286± 0.0286 +0.000000+0.000000 ±0.0032± 0.0032 Transolver 0.0069810.006981 ±0.0007± 0.0007 0.1428720.142872 ±0.0575± 0.0575 0.0057120.005712 ±0.0024± 0.0024 0.0087370.008737 ±0.0028± 0.0028 +0.000000+0.000000 ±0.0007± 0.0007 Bulk Only 0.0073660.007366 ±0.0007± 0.0007 0.1210420.121042 ±0.0188± 0.0188 0.0057710.005771 ±0.0018± 0.0018 0.0074950.007495 ±0.0020± 0.0020 +0.000241+0.000241 ±0.0007± 0.0007 Finite Support 0.0062400.006240 ±0.0009± 0.0009 0.1134060.113406 ±0.0413± 0.0413 0.0058550.005855 ±0.0019± 0.0019 0.0063890.006389 ±0.0022± 0.0022 â0.000150-0.000150 ±0.0007± 0.0007 TIDE 0.0061130.006113 ±0.0005± 0.0005 0.1109040.110904 ±0.0048± 0.0048 0.0056600.005660 ±0.0014± 0.0014 0.0062790.006279 ±0.0020± 0.0020 â0.000248-0.000248 ±0.0009± 0.0009 TIDES 0.0057460.005746 ±0.0005± 0.0005 0.1043970.104397 ±0.0036± 0.0036 0.0055810.005581 ±0.0013± 0.0013 0.0058260.005826 ±0.0018± 0.0018 â0.057067-0.057067 ±0.0114± 0.0114 Appendix G Additional OOD and Long-Horizon Results Tables 10, 11, and 12 report the numeric OOD and horizon-extension results referenced in Section 4.7. We place these tables in the appendix to keep the main experimental narrative compact while preserving the full metric breakdown. Table 10: OOD results on motion/trajectory OOD. Lower is better. Model lrollâl_roll târâaâjendâtraj_end nearânear farâfar Îdiv _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 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. Model lrollâl_roll târâaâjendâtraj_end nearânear farâfar Îdiv _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 TIDE 0.006519 0.690246 0.005904 0.006485 -0.000220 TIDES 0.006018 0.684433 0.005857 0.006022 -0.056036 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. Step Model lrollâl_roll nearânear farâfar Îdiv _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 24 3D ResUNet 0.011981 0.007319 0.015942 +0.000000 U-NO 0.006826 0.006991 0.005877 -0.000267 GNS 8.714556 8.704857 8.918174 +0.000000 Transolver 0.010734 0.006887 0.013558 +0.000000 TIDE 0.006489 0.006513 0.006105 -0.000071 TIDES 0.006089 0.006219 0.005744 -0.045924 Figure 4 visualizes long-horizon velocity-error growth, complementing the numeric results in Table 12. Figure 5 shows representative size-OOD examples used to evaluate object-scale generalization. Figure 4: Long-horizon velocity error growth on the test split. Figure 5: Representative size-OOD dataset examples used for object-scale generalization evaluation. Appendix H Additional Controlled Baseline Comparisons H.1 Parameter-Matched Localized U-NO Control To separate the effect of localized conditioning from the additional TIDE/TIDES mechanisms, we introduce a direct-localized U-NO control. It receives the same solid occupancy, boundary velocity, and derived interface representation, but does not use the TIDE mediator or the TIDES transport and correction modules. This experiment uses the same train/validation/test splits, target construction, normalization, rollout definition, and evaluator as Table 2. However, it uses a separate training protocol. The original main comparison uses a fixed optimization budget across model families. For this additional control, we adjust the U-NO width to match TIDEâs approximately 338M-parameter budget and retrain all four models with a common extended optimization schedule and five paired random seeds. The longer schedule reduces the risk that the modified U-NO configurations are disadvantaged by insufficient optimization. TIDE and TIDES retain their original parameter counts but are retrained under the same extended schedule. Consequently, the means in Table 13 are comparable within this additional experiment and should be distinguished from the original fixed-budget results in Table 2. 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 Params. lrolll_roll târâaâjendtraj_end lrollH=24l_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 Direct localized inputs improve U-NOâs mean rollout error from 0.00570 to 0.00548. Relative to this localized control, TIDE reduces mean rollout error by 3.3% and trajectory endpoint error by 9.7%. TIDES further reduces mean rollout error by 5.3% relative to TIDE. These results indicate that localized inputs account for part of the improvement, while the additional mediation and transport/stabilization mechanisms provide further mean gains. The absolute localized-U-NO-to-TIDE rollout difference is 0.00018. We interpret this as a modest mean improvement and do not claim statistical significance from the aggregate means alone. For paired seed comparisons, the relevant quantity is dj=lroll,jlocalizedâUâ-âNOâlroll,jTIDE,j=1,âŠ,5,d_j=l_roll,j^localized\ U -NO-l_roll,j^TIDE, j=1,âŠ,5, where a positive djd_j indicates lower TIDE error for the same seed. H.2 Shared Boundary and Physical-Loss Controls We additionally test whether unequal access to boundary enforcement or output-level physical losses explains the model ordering. The same hard boundary overwrite is applied to U-NO, TIDE, and TIDES, using the operator defined in Appendix E.4. A separate training control applies the same applicable output-level physical losses across the three models. Table 14 reports the resulting rollout errors. Each additional setting improves the reported mean error, while TIDES retains the lowest value. Under the shared physical-loss setting, its error is 5.4% below TIDE and 10.4% below U-NO. Table 14: Rollout error under shared boundary enforcement and physical-loss controls. The reference column uses the additional extended-schedule runs. 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 I Target and Solid-Conditioning Sensitivity I.1 Particle-to-Grid Target Sensitivity The dense target is constructed from particle simulations according to Appendix D. To assess dependence on this construction, we evaluate the same fixed model predictions against alternative targets and spatial supports: nearest-support filling, SPH-kernel interpolation, kernel-supported voxels only, and a smoothed dense target. Table 15 shows that the absolute error scale changes, while the reported ordering remains unchanged. TIDES has approximately 8.3â8.4% lower error than localized U-NO across these four evaluations. Direct particle-hit voxels occupy 4.2% of the evaluated liquid grid, whereas 93.8% of liquid voxels lie within the SPH kernel support of at least one particle. The remaining 6.2% lie outside that kernel support. Direct particle hits, kernel support, and the CIC support mask used for target construction are distinct support definitions and should not be conflated. These checks hold predictions fixed; they do not evaluate retraining under alternative target constructions. The preserved ordering reduces concern that nearest-support filling alone produces the comparison, but does not establish that the constructed target is an exact continuum velocity field. Table 15: Target-construction sensitivity using fixed predictions. Lower rollout error is better. 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 I.2 Sensitivity to Future Solid-State Errors We perturb the supplied future solid-state sequence with temporally smooth trajectory deviations while retaining the original reference liquid rollout. Perturbation magnitude is reported as the realized RMS solid-position discrepancy in voxel units. Table 16 measures how inaccurate conditioning affects TIDES. Rollout error increases by 1.6% at a 0.40-voxel RMS discrepancy and 13.3% at 1.60 voxels. The result characterizes robustness to conditioning error; it does not establish performance under independently simulated prescribed trajectories or closed-loop fluidâsolid coupling. Here, târâaâjendtraj_end measures forecast liquid-trajectory endpoints, rather than the error in the solid trajectory. 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. RMS error (voxels) lrolll_roll Interface error târâaâjendtraj_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 J Additional Temporal and Physical Diagnostics The following evaluations use free-running autoregressive predictions after training with an eight-step horizon. For notation, let t denote the observed starting time and k the forecast offset: k=1,âŠ,Hk=1,âŠ,H. Only the initial liquid state is observed; subsequent liquid inputs come from the modelâs predictions, while the future solid-state sequence remains supplied. J.1 Per-Step Interface and Far-Field Error We evaluate U-NO, TIDE, and TIDES on the same 64 in-distribution test windows. To define regions independently of the filled velocity target, let Ot+kO_t+k be the simulator solid-occupancy mask, Mt+kM_t+k the valid liquid mask, and rD_r voxel dilation with radius r. The diagnostic regions are Mt+kint=Mt+kâ©4â(Ot+k),Mt+kfar=Mt+kâ8â(Ot+k).M_t+k^int=M_t+k _4(O_t+k), M_t+k^far=M_t+k _8(O_t+k). The intermediate four-to-eight-voxel band is excluded from these two diagnostics. These masks differ from the original distance-based near/far definitions in Appendix D; the resulting numbers should therefore be interpreted separately. Table 17 reports selected points from the 24-step evaluation. Errors need not increase monotonically because motion magnitude and interface location vary over time. At offset 24, TIDES reduces interface and far-field errors by 5.4% and 9.6%, respectively, relative to TIDE. Table 17: Selected free-running interface/far-field errors. Each cell reports interface error / far-field error under the geometry-derived diagnostic masks. Model t+1t+1 t+8t+8 t+16t+16 t+24t+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 J.2 Temporal-Change and Velocity-Derived Diagnostics Let âšâ â©M · _M denote averaging over the common valid spatial mask, including component averaging for vector quantities. We set u^t=ut u_t=u_t so that the first predicted temporal increment is measured relative to the shared observed state. Temporal-change error is Echange=1Hââk=1Hâš|(u^t+kâu^t+kâ1)â(ut+kâut+kâ1)|â©M.E_change= 1H _k=1^H |( u_t+k- u_t+k-1)-(u_t+k-u_t+k-1) | _M. This is an error in frame-to-frame velocity increments. Additional diagnostics compare divergence, vorticity, and kinetic energy derived from the velocity fields: Ediv E_div =1Hââk=1Hâš|âgâ u^t+kââgâ ut+k|â©M, = 1H _k=1^H | _g· u_t+k- _g· u_t+k | _M, Dabs D_abs =1Hââk=1Hâš|âgâ u^t+k|â©M, = 1H _k=1^H | _g· u_t+k | _M, Evort E_vort =1Hââk=1Hâš|âgĂu^t+kââgĂut+k|â©M, = 1H _k=1^H | _gĂ u_t+k- _gĂ u_t+k | _M, Eenergy E_energy =1Hâk=1Hâš|q(u^t+k)âq(ut+k)|â©M,q(u)=12âc=13uc2. = 1H _k=1^H |q( u_t+k)-q(u_t+k) | _M, q(u)= 12 _c=1^3u_c^2. Here, âg _g denotes the common discrete spatial operator. The energy diagnostic uses unit-density kinetic energy in the evaluation coordinates. All models use identical masks and spatial operators. Table 18 reports means over offsets 1â24. TIDES has the lowest reported value in all five columns. In particular, its mean absolute divergence is 37.2% below TIDE. The accompanying vorticity and energy errors provide complementary evidence when interpreting divergence reduction, although they do not establish exact conservation or rule out every form of damping. Unlike Îdiv _div, which measures a change across correction stages, DabsD_abs measures absolute divergence of the final prediction. Neither quantity alone establishes physical fidelity. Pressure and free-surface state are not common prediction targets across the compared models, so these diagnostics do not constitute a complete physical evaluation. Table 18: Additional diagnostics averaged over free-running offsets 1â24. Lower values are better. Model EchangeE_change EdivE_div DabsD_abs EvortE_vort EenergyE_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 J.3 Free-Running Evaluation at Horizons 48 and 64 Table 19 extends evaluation to horizons 48 and 64. It supplements the original horizon-16/24 results in Table 12; the original fixed-budget results remain identified by their earlier protocol. At horizon 64, TIDES has 11.0% lower rollout error than TIDE and 18.4% lower error than U-NO. Its absolute divergence at offset 64 is 36.4% below TIDE and 44.7% below U-NO. The measurements show gradual error growth over the evaluated horizons, rather than abrupt failure. They do not establish stability at arbitrarily long horizons. 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 lrollH=48l_roll^H=48 lrollH=64l_roll^H=64 Dabs,48D_abs,48 Dabs,64D_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 Table 20 further characterizes the spatial distribution of absolute divergence at offset 64. Both the regional mean and the 95th percentile are lower for TIDES near the interface and in the far field. This provides a regional view of stabilization beyond a single global average. Table 20: Regional absolute-divergence statistics at offset 64. Each cell reports mean / 95th percentile. 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 J.4 Advected-Tracer Volume Diagnostic To examine volume behavior under predicted transport, we advect a common initial tracer set and reconstruct its occupied volume. Every model uses identical tracer initialization, interpolation, advection, and reconstruction. Table 21 reports the resulting relative volume drift. TIDES reaches 2.8% drift at offset 64, compared with 4.8% for TIDE and 5.5% for U-NO. This diagnostic measures volume behavior induced by velocity-based tracer transport. It depends on the shared reconstruction procedure and is not a direct measurement of exact mass conservation or predicted free-surface geometry. Table 21: Relative volume drift from shared advected-tracer reconstruction. Lower drift is better. 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 K Shape and Rotation Breakdown We disaggregate the additional evaluation by object family and solid angular motion. Table 22 reports sphere, cube, and cone results, together with low-, medium-, and high-rotation groups. TIDES has the lowest reported mean rollout error in every group. Its shape-wise improvement over TIDE is approximately 5.0â5.5%, and its improvement across the rotation groups is approximately 5.3â5.6%. Thus, the aggregate ordering is not confined to the sphere family or to low-rotation trajectories. These comparisons concern the benchmarkâs three existing families and do not establish generalization to unseen non-convex geometries. Given a complete solid geometry and boundary-velocity sequence, solid density affects the conditional liquid problem through the resulting solid motion when rigid-body dynamics are solved jointly. Density-conditioned prediction of the solid trajectory is a separate extension requiring joint fluidâsolid state updates. Table 22: Additional rollout-error breakdown by object family and solid rotation group. Lower 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 L Comparison with Simulation at Matched Runtime The latency comparison in Appendix C uses the full DualSPHysics configuration as a reference. To additionally compare accuracy at similar runtime, we evaluate coarse SPH on the same scenes and hardware, with its runtime matched to the TIDES inference budget. Table 23 reports runtime per physical second and mean error against the high-fidelity benchmark target. TIDES requires 0.3029 s, compared with 0.3015 s for coarse SPH, and has 30.8% lower mean error over the matched evaluation set. The full DualSPHysics configuration defines the reference target; its self-error is therefore not treated as a predictive result. At the benchmarkâs 8 Hz sampling rate, an eight-step query covers one physical second. TIDES produces approximately 26.4 output steps/s, corresponding to 3.30 times this sampling rate. Its latency is approximately 21.3 times lower than the full DualSPHysics reference. These ratios apply to the reported scenes, hardware, output target, and solver configurations. The comparison does not establish superiority over all real-time SPH or XPBD implementations. Appendix M Supplementary Protocol and Release Scope M.1 Video Evaluation Protocol The supplementary MP4 comparisons contain the reference sequence and outputs from every reported model. These videos use teacher-forced one-step prediction: each predicted frame receives the preceding reference liquid state, u^t+koneâ-âstep=fΞâ(ut+kâ1,st+k). u_t+k^one -step=f_Ξ(u_t+k-1,s_t+k). They visualize individual temporal transitions without accumulating predicted-input errors. In free-running autoregressive evaluation, the preceding prediction is used instead, u^t+krollout=fΞâ(u^t+kâ1rollout,st+k),u^trollout=ut. u_t+k^rollout=f_Ξ( u_t+k-1^rollout,s_t+k), u_t^rollout=u_t. The quantitative temporal and long-horizon diagnostics in Appendix J follow this free-running protocol. For recurrent models, the equations abbreviate the accompanying latent-state updates. Table 23: Accuracy comparison at matched runtime. Error is averaged over the matched evaluation set against the high-fidelity target. 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 M.2 Reproducibility and Release Materials We commit to releasing the DualSPHysics scene-generation scripts and simulator configurations; raw fluid-particle trajectories and recorded solid states; particle-to-grid conversion and dense-target construction; train/validation/test and OOD split manifests; forecasting and diagnostic evaluation code; adapted baseline implementations; TIDE/TIDES code and checkpoints; and environment and reproduction scripts. The release will distinguish the original fixed-budget experiments from the additional extended-schedule controls. Reproduction records will identify seed values, optimization schedules, checkpoint-selection rules, evaluated windows, normalization statistics, software versions, and dataset checksums. The supplementary package includes dataset examples and the scripts and environment specifications needed to inspect the generation and forecasting pipeline. M.3 Conditional Task Boundary As described in Appendix B, the simulator generates fluid-coupled rigid-body motion, and the realized future solid-state sequence is then supplied to the forecaster. The learned model predicts the conditional liquid response and does not update the solid trajectory through predicted hydrodynamic forces or torques. Joint fluidâsolid stability is therefore outside the present evaluation. The additional controls strengthen the evidence within this conditional benchmark. Its synthetic data, restricted geometry and physical coverage, known-future-solid assumption, and benchmark-defined dense target remain limitations. Extensions to uncertain motion, non-convex objects, multiple solids, broader material parameters, real observations, and closed-loop two-way coupling require further validation. NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paperâs contributions and scope? Answer: [Yes] Justification: The abstract and introduction state the task formulation, benchmark, TIDE/TIDES framework, and empirical scope. The claims are supported by the main comparison, controlled ablations, OOD tests, and long-horizon evaluation in Sections 4â7. 2. Limitations Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: Section 6 explicitly discusses the controlled nature of the benchmark, the use of synthetic data, the limited OOD axes, and the fact that the dense Eulerian target is constructed from particle simulations rather than measured as a solver-native continuum field. 3. Theory assumptions and proofs Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? Answer: [N/A] Justification: The paper does not present formal theoretical results, theorems, or proofs. The main contributions are a task formulation, benchmark, model framework, and empirical evaluation. 4. Experimental result reproducibility Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)? Answer: [Yes] Justification: The paper provides the task setup, data splits, evaluation metrics, baseline adaptation protocol, model structure, loss decomposition, and seed-stability results in the main text and appendices. Simulator settings, particle-to-grid conversion rules, and implementation details are also described in the appendix. 5. Open access to data and code Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? Answer: [Yes] Justification: The data generation pipeline, benchmark protocol, model implementation, and evaluation scripts will be released with sufficient documentation to reproduce the main experimental results. The released package will include instructions for preparing the benchmark data, training the proposed models and adapted baselines, and computing the reported metrics. 6. Experimental setting/details Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer) necessary to understand the results? Answer: [Yes] Justification: Section 4 specifies the dataset split, grid resolution, rollout horizon, OOD settings, evaluation metrics, and shared comparison protocol. Additional implementation details, tensor shapes, loss weights, baseline adaptations, and metric definitions are provided in the appendix. 7. Experiment statistical significance Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [Yes] Justification: The main comparison reports mean performance over five random seeds, and Appendix F reports the corresponding mean and standard deviation for the main metrics. The reported variability reflects different random seeds under the same benchmark protocol. 8. Experiments compute resources Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments? Answer: [Yes] Justification: The paper reports the shared optimization budget and rollout latency for the proposed method and baselines, and Appendix C provides a controlled latency comparison against the reference simulator. The released code and documentation will include the compute environment used for reproducing the experiments. 9. Code of ethics Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.c/public/EthicsGuidelines? Answer: [Yes] Justification: The research uses synthetic physical simulation data and does not involve human subjects, private data, or sensitive personal information. The paper is anonymized for review and is intended as a scientific benchmark and modeling study. 10. Broader impacts Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? Answer: [Yes] Justification: The paper motivates positive uses in robotic liquid manipulation, digital twins, simulation-driven design, and rigidâfluid content generation. It also limits its claims by emphasizing that the benchmark is a controlled forecasting study rather than a replacement for physical simulation, which helps reduce the risk of overclaiming deployment readiness. 11. Safeguards Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pre-trained language models, image generators, or scraped datasets)? Answer: [N/A] Justification: The paper does not release high-risk models such as language models, image generators, or scraped human datasets. The assets concern synthetic physical simulation and liquid-forecasting benchmarks, so the listed high-risk release safeguards are not directly applicable. 12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [Yes] Justification: The paper identifies the use of DualSPHysics 5.4 for dataset generation and cites the relevant external model families and baselines. The released code and documentation will include asset versions and license information for simulator and software dependencies. 13. New assets Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets? Answer: [Yes] Justification: The paper introduces a new benchmark and documents the task definition, data construction, splits, metrics, particle-to-grid conversion, baseline adaptation, and limitations in the main text and appendix. The released assets will include documentation describing data format, generation protocol, evaluation scripts, and intended use. 14. Crowdsourcing and research with human subjects Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)? Answer: [N/A] Justification: The paper does not involve crowdsourcing, user studies, or research with human subjects. All experiments are conducted on synthetic simulation data. 15. Institutional review board (IRB) approvals or equivalent for research with human subjects Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained? Answer: [N/A] Justification: The paper does not involve human subjects, personal data, or crowdsourced participants. Therefore IRB approval or equivalent human-subjects review is not applicable. 16. Declaration of LLM usage Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigor, or originality of the research, declaration is not required. Answer: [N/A] Justification: The core methodology, data generation, model training, and evaluation do not use LLMs as an important, original, or non-standard component. Any writing or formatting assistance, if used, does not affect the scientific method or experimental results.