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Energy-Structured Latent World Models with Neural Time Fields for Physically Constistent Open-World Motion Planning
Yapeng Liu, Yuanzhao Zhai, Bo Ding, Huaimin Wang, Lin Wang
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Abstract
Abstract:Physically consistent motion planning remains a fundamental challenge in embodied AI, as generated trajectories must strictly conform to real-world execution dynamics. While latent world models offer a promising approach by predicting these dynamics, existing methods learn unconstrained future representations where absorbed physics remains implicit. Therefore, they fail to form reusable physical knowledge, which compromises reliability in unpredictable open-world navigation. To address this, we propose a novel Energy-Structured Latent World Model (ELWM). Our key idea is to structure the ELWM latent state to explicitly carry energy and momentum, ensuring strictly causal transitions via dissipation and control ports. Trained on multimodal RGB-D and inertial interaction histories, our model guarantees physically consistent predictions. We further implement this for motion planning by constructing Physics-Conditioned Neural Time Fields (PC-NTF), a key technical cornerstone that integrates ELWM into an arrival time field via the Eikonal equation to yield a physically-informed navigation policy. Across held-out scenes, our evaluation reveals significant improvements. Compared to generic latent models, PC-NTF reduces 0.8-s motion-prediction NRMSE from 0.36 to 0.29. Against Active Neural Time Fields, it improves navigation success from 81.3% to 89.7% and SPL from 0.64 to 0.73, while cutting the physical collision rate from 12.1% to 5.8% and the Eikonal residual from 0.083 to 0.031. Beyond these targeted gains, our results demonstrate that embedding explicit physical structures into latent spaces intrinsically bridges the gap between predictive world models and safe, dynamically feasible motion planning.
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- Source: https://arxiv.org/abs/2608.09876v1
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Energy-Structured Latent World Models with Neural Time Fields for Physically Constistent Open-World Motion Planning Yapeng Liu1,2,3, Yuanzhao Zhai1,2, Bo Ding1, Huaimin Wang1,2, Lin Wang3 Abstract Physically consistent motion planning remains a fundamental challenge in embodied AI, as generated trajectories must strictly conform to real-world execution dynamics. While latent world models offer a promising approach by predicting these dynamics, existing methods learn unconstrained future representations where absorbed physics remains implicit. Therefore, they fail to form reusable physical knowledge, which compromises reliability in unpredictable open-world navigation. To address this, we propose a novel Energy-Structured Latent World Model (ELWM). Our key idea is to structure the ELWM latent state to explicitly carry energy and momentum, ensuring strictly causal transitions via dissipation and control ports. Trained on multimodal RGB-D and inertial interaction histories, our model guarantees physically consistent predictions. We further implement this for motion planning by constructing Physics-Conditioned Neural Time Fields (PC-NTF), a key technical cornerstone that integrates ELWM into an arrival time field via the Eikonal equation to yield a physically-informed navigation policy. Across held-out scenes, our evaluation reveals significant improvements. Compared to generic latent models, PC-NTF reduces 0.8-s motion-prediction NRMSE from 0.36 to 0.29. Against Active Neural Time Fields, it improves navigation success from 81.3% to 89.7% and SPL from 0.64 to 0.73, while cutting the physical collision rate from 12.1% to 5.8% and the Eikonal residual from 0.083 to 0.031. Beyond these targeted gains, our results demonstrate that embedding explicit physical structures into latent spaces intrinsically bridges the gap between predictive world models and safe, dynamically feasible motion planning. Introduction Physically consistent motion planning is a fundamental challenge in embodied AI, the trajectory is useful only when the robot can execute it under its current dynamics (Hsu et al. 2002; Liu et al. 2025a). The collision-free geometric path may become unsafe or inefficient when dissipation payload, actuation limits, or the robot’s dynamic state changes, and these factors often have weak geometric signatures and become observable through interaction (Pham et al. 2013; Bohg et al. 2017; Liu et al. 2026). Physical consistent motion planning in open-world environment therefore requires a predictive model of how the robot–environment system responds to execution (Ha and Schmidhuber 2018; Kumar et al. 2021; Huang et al. 2026). Figure 1: ELWM–PC-NTF bridges action-conditioned physical prediction and motion planning. (A) Robot interactions need action–response capability, (B) ELWM predicts future motion and latent physical dynamics, and PC-NTF combines them with neural time fields. (C) System closed-loop execution produces obstacle-aware, physics-informed navigation towards the goal. Latent world models provide a promising basis for this predictive component (LeCun and others 2022; Bar et al. 2025). By encoding multi-modal observation and interaction histories into latent states, world models support action-conditioned rollouts without reconstructing every future observation (Hafner et al. 2019b, a, 2023). Recent models extend this principle to motion planning and navigation (Zhou et al. 2025; Chen et al. 2025). However, latent prediction does not imply reusable physical knowledge or formal physical consistency (Lutter et al. 2019; Shang et al. 2026). Existing methods mostly optimize unconstrained future representations, leaving dissipation, interaction response, and other physical regularities entangled in task-, env- and embodiment-specific transition weights (Ye et al. 2026; Gong et al. 2026; Chen et al. 2026). They expose neither a causal physical structure nor a predicted quantity that motion planner can generally consume. This limitation is especially consequential in open-world navigation, where a planner must adapt its decision to physical interactions that are not determined by scene geometry alone. To make physical dynamics reusable, we study the latent transition of how to learn transferable physical regularities and effectively implement them in motion planning. Consequently, we propose the Energy-Structured Latent World Model (ELWM), which encodes multimodal RGB-D and inertial interaction histories, together with executed trajectories, into the general physical regularities and dynamic latent phase state (Desai et al. 2021). We formalize the latent stored-energy function and structure the explict port-Hamiltonian (pH) transition for navigation dissipation and control ports (Zhong et al. 2019). This construction imposes an action-conditioned, energy-balance-compatible transition structure. Given a candidate command sequence, ELWM rolls this structured state forward and decodes planner-observable consequences, including the predicted pose, velocity, interaction, and traversal progress. The learned structure can therefore be evaluated through both latent future prediction and port-Hamiltonian motion planning dynamics. ELWM exposes explicit physical interaction consequences, while these predictions do not specify how the robot should progress toward a navigation goal by themselves. The motion planner requires a global value representation which connects locally achievable outcomes to global navigation cost (Maes et al. 2026). Neural time fields provide this complementary substrate by interpreting time-to-go as a continuous arrival-time function constrained by the Eikonal equation, and Active Neural Time Fields (ANTF) update the field from the observed geometry of the scene incrementally, the resulting arrival time and its goal-directed gradient efficiently organize global geometric guidance (Ni and Qureshi 2022; Liu et al. 2025b). However, ANTF derives this guidance primarily from static geometry map, it induces the same preference even when ELWM predicts different traversal responses under changes in payload or actuation. The missing link is therefore a motion planning system that utilizes ELWM observable, candidate-dependent predictions to condition arrival-time decisions, beyond the mappings from states, poses, or dissipation to an Eikonal propagation speed. We close this interface gap through Physics-Conditioned Neural Time Fields (PC-NTF) system. The overall motivation are summarized in Figure 1, ELWM and PC-NTF bridges the physical prediction and motion planning. PC-NTF decomposes motion planning responsibility between ELWM’s local interacting execution prediction and ANTF’s global geometric time-to-go envelope. ANTF first supplies the nominal descent direction and a shared set of feasible action candidates. Starting from the same interaction history, the trained ELWM predicts how each candidate will be executed over a short horizon. PC-NTF then queries the neural time field at each predicted terminal position and then fixes the physics conditioned traversal delay derived from the predicted interaction progress from a physics latent decoder. The resulting model-based time-to-go backup treats the world-model consequence as an action-conditioned physical stage cost and the neural time field as a geometric terminal value. It thereby connects ELWM to arrival-time planning without interpreting each latent mechanics as the Eikonal equation participant. Because the correction is candidate dependent online, PC-NTF can change the locally executed decision while preserving the global geometric structure of ANTF, without requiring ELWM to predict paths to the goal or the time field to be retrained for each physical interaction condition. We evaluate this framework along two complementary axes that mirror its two technical components. At the world-model level, ELWM is compared with a capacity-matched generic latent world model in MPC controller using observable multi-step motion NRMSE, action interventions, and Hamiltonian dynamics diagnostics on held-out physical regimes and scenes. At the planning level, PC-NTF is compared with geometry ANTF and we report navigation success, SPL, physical collision rate, arrival-time accuracy, Eikonal residual, and planning latency. This separation allows the contributions analysis of generic prediction, energy-structured dynamics, and neural time-field guidance to be measured independently. Experiments show that ELWM obtains motion NRMSEs of 0.7050.705 and 0.6970.697 on the held-out regime and scene, respectively, while maintaining a discrete energy-balance residual of 7.12×10−77.12× 10^-7; action shuffling increases latent prediction error by approximately 10%10\%. These results support action-conditioned prediction and consistency of the physical transition, whereas navigation positive improvement from success rate, latent error and Eikonal residual also indicates PC-NTF motion planning efficiency. Our contributions follows: • ELWM Construction. We introduce ELWM, an energy-structured latent world model that organizes action-conditioned dynamics through a latent phase state with explicit stored-energy, dissipation, and control structure. • PC-NTF System Implementation. We propose PC-NTF, the motion planning system that utilize ELWM predicted execution consequences with NTF geometric time-to-go through specific terminal states and physically dimensioned traversal delays. • Efficient motion planning results. We evaluate ELWM and PC-NTF through the physical consistent prediction and motion planning in open-world environments. Experiment shows that we reduce 0.8-s motion-prediction NRMSE from 0.36 to 0.29, improve navigation success from 81.3% to 89.7% and SPL from 0.64 to 0.73. Related Work Physically Consistent Latent World Models. Latent world models compress observation action-conditioned histories and future dynamics into meaningful representations, which can support planning and policy learning (Ha and Schmidhuber 2018; Hafner et al. 2019b, a, 2023). Recent models extend this principle to pretrained visual features, navigation, and robot policies (Zhou et al. 2025; Rao et al. 2026; Chen et al. 2026). These results establish the decision relevance of latent rollouts, but predictive sufficiency does not make robot–environment interaction mechanics reasonable: effects such as payload and dissipation may remain entangled in unconstrained latent transition (Nie et al. 2026; Schölkopf et al. 2021). Physical consistent world models were proposed to address this limitation, including physics-informed world models and mechanics-structured networks. The former incorporate differentiable simulators or physical supervision into predictive learning (Li et al. 2025; Shang et al. 2026), whereas the latter constrain the learned transition itself (Greydanus et al. 2019; Zhong et al. 2019). More recent approaches bring these principles closer to world models construction, AC-HGN embeds actions as external forces in an abstract Hamiltonian phase space (Troch et al. 2025); Phys-JEPA structures physical and residual components of latent prediction (Nie et al. 2026); and PH-Dreamer combines action-controlled energy flow and dissipation into recurrent world-model dynamics (Luan and Shi 2026). These advances establish that physical consistent structure can improve the latent prediction and control, while they mainly evaluate with in the learned model or policy. ELWM targets on the motion planning, extracting the navigation regularities and organize action-conditioned prediction through dissipation. In this way, ELWM provides the interaction dynamics and shows the efficiency on jointly motion planning. Neural Time Fields for Motion Planning. Efficient motion planning can be characterized by Hamilton–Jacobi and Eikonal equations, with classical solutions obtained by dynamic-programming and Fast Marching methods (Lions 2006; Sethian 1996). For motion planning, neural fields recently have emerged as suitable environment representations due to their compactness and continuity property (Park et al. 2019; You et al. 2023), and neural PDE solvers approximate Eikonal solutions continuously through data and equation residuals (Raissi et al. 2018; Smith et al. 2020; Bin Waheed et al. 2021). NTFields applies this formulation to robot configuration spaces without expert paths (Ni and Qureshi 2022); Active NTFields further learns arrival-time maps online in open-world environments (Liu et al. 2025b; Zhong et al. 2025); and Eikonal caging extends the formulation to contact-rich robot arm manipulation (Zhang et al. 2026). These methods define physical consistency primarily through the Eikonal PDE and geometry-derived propagation speed, but they did not take the dynamic online interaction consequences in consideration. PC-NTF lifts the arrival-time Eikonal formulation by using Practical physical traversal costs through its decoder and the physics condition, to form the online interactive motion planning framework. Open-World Motion Planning Under Changing Dynamics. Open-world motion planning requires navigation without fully known maps and fixed execution conditions (Du Toit and Burdick 2011). Classical sampling-based planners search for collision-free paths in a given configuration-space representation (Karaman and Frazzoli 2011), while learning-based planners accelerate this search using learned trajectory proposals and classical feasibility checks (Qureshi et al. 2020). These methods represent feasibility through geometric collision constraints. Further, experience-driven navigation complements geometric planning by learning local reachability or traversability from robot interaction. BADGR learns navigational affordances from self-supervised experience (Kahn et al. 2021) and WayFAST uses online traction estimates to supervise terrain-traversability prediction (Gasparino et al. 2022). However, their planner-facing quantities are typically expressed as collision. These evaluation do not explictly predict how each candidate command sequence will execute under the robot–environment setting. PC-NTF evaluates feasible motion candidates under the robot’s current interaction response along with ELWM, allowing open-world decisions to change with payload, dissipation, actuation limits, and dynamic state. Method Figure 2: Overview of ELWM. Physical history and interaction dynamics are encoded into physical and latent state; a controlled, dissipative port-Hamiltonian transition formalizes the prediction latent prior as motion futures P for PC-NTF planning (in Fig.3). ELWM Training by the state future consistency. Preliminary Problem Setting. We consider a differential-drive robot that navigates in a planar workspace Ω⊂ℝ2 ^2 toward the goal g in Eikonal form, its state can be expressed as t=(t,ψt,t),t=[vt,ωt],r_t= (x_t, _t, ξ_t ), ξ_t=[v_t, _t], (1) and the executed control is t=[vtcmd,ωtcmd] u_t=[v_t^cmd, _t^cmd]. An upstream navigation task module with VLM foundation model supplies the goal gx_g from dataset, μr _r denotes robot meta-information (mass/payload, velocity and acceleration limits). Actions at=(νt,Δt)a_t=( _t, t) are the velocity commands received by the base, and Htphy=oτphy,ξτ,aτ−1,μrτ=t−LtH_t^phy=\o_τ^phy, _τ,a_τ-1, _r\_τ=t-L^t collects the recent proprioceptive and navigation interaction history. Hypothesis. The physical quantity delivered to the planner is isotropic traversal-speed field: within one replanning window, the maximum speed the robot can stably sustain at each location is represented by scalar. Orientation-dependent friction, turning-radius limits, and acceleration state are absorbed into this scalar and enforced at execution time by an explicit constraint layer. For detailed neural time field calculation, we represent them inside through an anisotropic and state-space Hamilton–Jacobi formulation, which we analyze in PC-NTF realization (Bardi et al. 1997). System overview. PC-NTF couples a physical prediction world model with NTF planner through motion planning interface, the inference loop is summarized as t _t →ELWMt, \;ELWM\;s_t, (TA,t,) (T_A,r_t,g ) →NTFt(i)i=1N, \;NTF\; \U^(i)_t \_i=1^N, (t,t(i)i=1N,TA) (s_t, \U^(i)_t \_i=1^N,T_A ) →PC-NTFtexec⟶t+1. \;PC-NTF\; u^exec_t _t+1. ELWM summarizes the interaction history as ts_t; NTF uses the current state, goal, and arrival-time field TAT_A to generate shared candidates; and PC-NTF predicts their physical consequences, reranks them with the time field, and executes the selected feasible action. The observed transition then updates t+1H_t+1, closing the feedback loop. Energy-Structured Latent World Model Figure 2 demonstrates the ELWM overview from interaction latent encoding and pH-constrained motion prediction to PC-NTF. Latent physical state. The ELWM world model encoder EθE_θ mappings the interaction history to the latent physical state: (t,t,t)=Eθ(tphy),(M_t,Q_t,V_t)=E_θ(H_t^phy), (2) where tM_t captures the slow physical regularity dynamics (payload, ground response, actuation delay), and t,t∈ℝdz/2Q_t,V_t ^d_z/2 are latent generalized coordinates and velocities computed from relative motion features. The latent inertia G links velocity and momentum, and we parameterize it by Cholesky factor tc_t with robot meta-info r μ_r. t=θ(t,r,t)≻0,t=tt,G_t=G_θ(M_t, μ_r,c_t) 0, _t=G_tV_t, (3) We take t=[t,t]Z_t=[Q_t,P_t] as the port-Hamiltonian latent phase state, and the pairing in Eq. (3) supplies consistent velocity–momentum relation for the transition below. Latent stored energy. We formalize the latent stored energy as ℰθ(t,t;t,r)=12t⊤θ−1t⏟θ+Uθ((t),t,r)⏟θ,E_θ(Q_t,P_t;M_t, μ_r)= 12P_t G_θ^-1P_t_K_θ+ U_θ\! ( ρ(Q_t),M_t, μ_r )_U_θ, (4) with kinetic energy θK_θ and potential energy θU_θ. ρ(⋅)ρ(·) is a feature map restricted to relative quantities, which makes θU_θ invariant to global translations of tQ_t. Since kinetic parameterization gives ∂θ/∂t=t−1t=t _θ/ _t=G_t^-1P_t=V_t, the Hamiltonian flow of tQ_t coincides with the encoder’s latent velocity. Port-Hamiltonian transition. Navigation robot exchanges energy with environments: actuators inject work, and the interaction dissipates it. Following port-Hamiltonian formalism (Zhong et al. 2019; Desai et al. 2021), latent transition generated by the dissipation paradigm ˙t=(−θ(t,t,r))∇ℰθ+Bθ(t,t,r)t, Z_t= (J-R_θ(Z_t,M_t, μ_r) ) _ZE_θ+B_θ(Z_t,M_t, μ_r)\, u_t, (5) with skew-symmetric operator =[0−0]J= [ smallmatrix0&I\\ -I&0 smallmatrix ], dissipation θ⪰0R_θ 0, control port BθB_θ, and latent control input t=Eu(t,t,r) u_t=E_u( a_t, o_t, μ_r) mapping velocity and actuator commands. The ELWM physical transition structural property follows directly, defining the port output t=Bθ⊤∇ℰθ y_t=B_θ _ZE_θ and using ⊤=−J =-J, dℰθdt=∇ℰθ⊤˙t=−∇ℰθ⊤θ∇ℰθ+t⊤t. dE_θdt= _ZE_θ Z_t=-\, _ZE_θ R_θ _ZE_θ+ u_t y_t. (6) Latent energy changes is the injected system latent work minus a non-negative dissipated power, and it is conserved when ut=0u_t=0 and Rθ=0R_θ=0. Equation (6) holds as an identity of the vector field in Eq. (5) for energy balance; acceleration and friction are explained by the control port and dissipation terms. For momentum, the pH transition gives ˙t=−∂ℰθ/∂t+[Bθt−θ∇ℰθ] P_t=- _θ/ _t+[B_θ u_t-R_θ _ZE_θ]_P; when the Hamiltonian is invariant along a latent direction and that direction is unforced and dissipation-free, the corresponding momentum component is constant. Future predictions of latent states are produced by integrator ΦΔtpH ^pH_ t. Since GθG_θ depends on QtQ_t, the energy in Eq. (4) is non-separable, we use the implicit midpoint as discrete-gradient updates for dissipation. Training. In our design, physical regularities are carried by the transition structure, so the world model training task reduces to future prediction. Targets are encodings of future histories by the same encoder, Zt+k+=Eθ(Ht+kphy)Z_t+k^+=E_θ(H_t+k^phy)_Z, and the objective is ℒELWM=1K∑k=1Kwkdz‖Z^t+k−Zt+k+‖22+λsigℛSIGReg,L_ELWM= 1K _k=1^K w_kd_z \| Z_t+k-Z_t+k^+ \|_2^2+ _sig\,R_SIGReg, (7) the task-level future-latent prediction objective with representation-geometry regularizer: ℛSIGRegR_SIGReg is the isotropic-Gaussian embedding regularizer of LeJEPA (Balestriero and LeCun 2025) applied to current, target, and rolled-out latents, which prevents collapse. Gradients propagate through the structure-preserving rollout into ℰθ,Rθ,Bθ,EuE_θ,R_θ,B_θ,E_u, we control the port-Hamiltonian mechanism constraints make effects that the model can reduce prediction error only after the physics. Energy residuals εenergy=(Δℰ^t−W^t)2 _energy=( E_t- W_t)^2 and symmetry-masked momentum residuals ar e logged for checkpoint selection; reinstating them in the loss is evaluated as an ablation. After convergence the world model exposes the physical context and world state for PC-NTF navigation planning. PC-NTF System Construction Figure 3: PC-NTF motion planning pipeline. (a) NTF proposes shared motion candidates -> (b) ELWM predicts their action-conditioned execution consequences -> (c) PC-NTF selects the candidate with the most efficient traversal delay. PC-NTF implements the ELWM priors with neural time fields for complete navigation system, ELWM supplies implicit physics, and the Eikonal solver formalize the time fields initialization. Figure 3 illustrates how PC-NTF reranks NTF candidates using ELWM-predicted motion consequences in sequence. We build PC-NTF in three steps—latent physics decoder, condition time fields, and the generalized time-field problem(Liu et al. 2025b): sgeo,t()‖∇T‖2=1⏟geometry Eikonal⟶seff,t(;ζt(),μr)‖∇T‖2=1⏟physics-conditioned Eikonal, s_geo,t(x)\,\| _xT\|_2=1_geometry Eikonal\;\; \;\; s_eff,t (x;\, _t(x), _r )\,\| _xT\|_2=1_physics-conditioned Eikonal, (8) where ζt(x)=Emap(bt,x) _t(x)=E_map(b_t,x) denotes map construction at the motion planning. Equation (8) generalizes the solver objective, and additional dependency (ζt _t, r μ_r) is available before planning begins. Physics latent decoder. Learning form the ELWM future prediction context, we take it as the action-conditioned consequences of motion planning. To better use the physical latent prediction for motion planning, we design the PC-NTF physics latent decoder which maps latent into traversal speed, and this decoder is trained by Huber loss from the physics-speed envelopes t∗s^*_t. Given the interaction history tH_t and a candidate command sequence t(i)U^(i)_t, the decoder exposes planner-facing representation t(i)=WM(t,t(i))=(^t(i),^t+H(i),ΔTWM(i),t(i)), ^(i)_t=A_WM\! (H_t,U^(i)_t )= ( τ^(i)_t,\, x^(i)_t+H,\, T^(i)_WM,\, ρ^(i)_t ), (9) where ^t(i) τ^(i)_t is the decoded observable motion, ^t+H(i) x^(i)_t+H is its terminal state, ΔTWM(i)≥0 T^(i)_WM≥ 0 is traversal delay, and t(i) ρ^(i)_t summarizes physical predicted tracking, safety, and control consequences. NTF Construction and Physics Condition. NTF represents scene by a continuous arrival-time field T(,)T(x,g), which estimates the remaining travel time from a location x to the goal gx_g. Its geometry-derived speed S()S(x) and arrival time follow the Eikonal relation S()‖∇T(,g)‖2=1,T(,)=0.S(x) ∇ T(x,x_g) _2=1, T(g,g)=0. (10) Time field is parameterized by PINN neural network and fitted from location–goal pairs using the Eikonal equation and the zero-time boundary condition at the goal. As new geometric observations become available, Active NTF updates the clearance-dependent speed supervision and the corresponding arrival-time representation without requiring expert trajectories (Ni and Qureshi 2022). Regions with greater obstacle clearance admit faster propagation and therefore smaller arrival time, while following the negative field gradient provides a goal-directed motion cue. This field captures scene geometry efficiently in navigation, but it cannot by itself distinguish physical changes in dissipation, actuation, or motion response. PC-NTF complements the geometric field with action-conditioned predictions from the ELWM. The NTF descent direction first defines a nominal motion, from which the planner forms a small set of feasible alternatives. ELWM rolls out every alternative from the same interaction history and decodes its predicted pose and velocity. These predictions P go through “pH” and decoded by the physics latent decoder. determine two quantities that are directly meaningful to the time field: the location that the robot is expected to reach and any additional time caused by slower-than-commanded progress. For candidate i, PC-NTF evaluates T(i)=T(^t+H(i),g)+ΔTphys(i).T^(i)=T\! ( x^(i)_t+H,x_g )+ T^(i)_phys. (11) Neural time fields solve the Eikonal equation, T(^t+H(i),g)T\! ( x^(i)_t+H,x_g ) is the geometry time remaining from the predicted terminal location. ELWM provide the Tphys(i)T^(i)_phys after decoder, representing the physical regularities and predicted motion by the world model along with the navigation progress. Consequently, a geometrically attractive motion receives a larger cost when the current physical regime makes it difficult to realize. PC-NTF therefore physically conditions the use of the neural time field without treating the latent state, mechanical energy, or dissipation matrix as an Eikonal speed. Motion-planning and navigation system. At every replanning step, the system follows one execution order: query the NTF field, generate shared feasible candidates, predict their motion with the world model, and evaluate them using Eq. (11). Candidates with poor tracking, speed-limit violations, high uncertainty, or abrupt control changes are penalized, and the first action of the lowest-cost feasible candidate is executed. The resulting pose, velocity, and executed action are then appended to the interaction history before the next planning step. The ELWM and NTF are trained separately from interaction trajectories and scene geometry, respectively, and remain frozen during system evaluation. This construction preserves NTF as the source of global geometric guidance while using the world model to supply the physical consequences that a geometry-only time field cannot observe. Equation (8) is also the mechanism by which the world model influences planning: when the readout reports reduced capability—higher dissipation, heavier payload—seff,ts_eff,t drops, the Eikonal constraint forces larger time gradients, the same Euclidean distance costs more arrival time, and planned paths bend away from physically slow regions without any explicit penalty. Motion generation follows the solved field: texec=ΠCt[−seff,t()∇T(;g)‖∇T(;g)‖2+ϵ], ^exec_t= _C_t [-s_eff,t(x)\, ∇ T(x;x_g)\|∇ T(x;x_g)\|_2+ε ], (12) where the implied speed s^t=1/(‖∇xTt‖2+ϵ)≈seff,t s_t=1/(\| _xT_t\|_2+ε)≈ s_eff,t serves as a built-in diagnostic of how faithfully the solver absorbed the physical prior, and ΠCt _C_t is a deterministic projection enforcing occupancy boundaries, unknown-space rules, velocity and acceleration clipping, and emergency stops. Experiments Experimental Setup Environments. (i) iGibson indoor scenes (Shen et al. 2020) with cluttered multi-room layouts; (i) held-out environments sampled from HM3D (Ramakrishnan et al. 2021) for topological complexity, formalize into iGibson format; (i) a terrain-variation suite with heterogeneous ground surfaces (carpet, tile, gravel) and payload changes that alter interaction dynamics. Train/test splits are by scene, surface, and robot meta-configuration; adjacent frames have bounding split. Baselines. The main comparison includes Active Neural Time Fields(Liu et al. 2025b), Generic world models (the capacity matched generative action-consistent world model)(Chen et al. 2026) with the MPC budget, and the complete ELWM+PC-NTF system. Comparison axes. We organize the evaluation following two axes. At the world-model axis, we ask whether imposing an energy-structured transition improves action-conditioned prediction over a capacity-matched unconstrained latent world model. ELWM is compared with a capacity-matched Generic WM (Chen et al. 2026). At the planning system axis, we ask how these predictions change the motions selected from the same ANTF candidate set, PC-NTF is compared with geometry ANTF (Liu et al. 2025b). Metrics. The world-model axis uses 0.8-s motion-prediction normalized root mean square error (NRMSE@0.8s; lower is better), pose error, and twist error. The planning axis uses success rate, success weighted by path length (SPL), physical collision rate, and Eikonal residual. Experimental Results Physical Consistent World Model Prediction. Table 1 isolates the predictive contribution of the energy-structured transition. ELWM reduces observable-motion NRMSE from 0.360.36 to 0.290.29 relative to Generic WM, an 19.4%19.4\% reduction. Pose and twist errors respectively decrease by 10.8%10.8\% and 1.8%1.8\%. Along with the experiment, a larger action-intervention ratio (1.1931.193 to 1.0091.009) shows that the ELWM rollout depends more strongly on the executed control. Also, there is a smaller positive difference in slow-M intervention (1.0621.062 to 1.0331.033), showing that MtM_t has the potential to identify reusable physical regularities. Method NRMSE @0.8s↓ Pose↓ Twist↓ Action int.↑ Generic WM 0.36 0.749 0.489 1.009 ELWM 0.29 0.669 0.480 1.193 Table 1: World-model physical consistency analysis. Prediction errors are evaluated on paired cases under identical observations, action constraints, and candidate motions in indoor scenes. Physical world model analysis includes energy consistency, our trained ELWM checkpoint attains an energy-balance residual of 1.88×10−71.88× 10^-7, and its predicted dissipated power is nonnegative on all evaluated rollout steps, verified that the numerical transition respects the imposed dissipative pH identity and ELWM energy-structured model. Metrics Setting Regime↓ Scene↓ Motion NRMSE Random init. 7.753 7.699 !9ELWM !9 0.705 ↓ 90.9% \,90.9\% !9 0.697 ↓ 91.0% \,91.0\% Latent MSE Random init. 0.1695 0.1656 !9ELWM !9 0.0856 ↓ 49.5% \,49.5\% !9 0.0835 ↓ 49.6% \,49.6\% Table 2: Comparison with random initialization, showing ELWM learning effects and generalization on held-out cases. ELWM substantially improves prediction on both held-out axes, motion NRMSE decreases from 7.753 to 0.705 on unseen physical regimes and from 7.699 to 0.697 on unseen scenes, corresponding to relative reductions of 90.9% and 91.0%. Latent MSE is also reduced by 49.5% and 49.6%. The closely matched reductions show that ELWM training process effectively improves both held-out tests on hold out. Motion Planning and Navigation. Table 3 shows the motion planning systems comparison, and Figure 4 visualizes the matched planning comparison. For efficient and physical motion planning, PC-NTF is intended to improve physics-awared navigation because ELWM supplies a physics-conditioned estimation of how the current system will respond to navigation interaction based on learned physical regularities. This prior modifies the effective propagation cost and penalizes candidates that are trajectory valid but physically slow or difficult to execute, PC-NTF formalize them into Neural Time Fields for motion planning. Compared with ANTF, PC-NTF improves SR from 81.3% to 89.7%, SPL from 0.64 to 0.73, reduce Eikonal residual from 0.083 to 0.031. ELWM physics prior bring more latency than origin ANTF, while we expose that PC-NTF latency is better than MPC control port (371.7 to 193.2). Figure 4: Motion-planning and navigation results comparison, PC-NTF indicates ELWM+PC-NTF system. Method SR (%)↑ SPL↑ Phys. Coll. (%)↓ Eik.↓ Latency (ms)↓ ANTF 81.3 0.64 14.4 0.083 35.3 Generic WM + MPC 87.1 0.74 6.5 0.143 371.7 ELWM + PC-NTF 89.7 0.73 5.8 0.031 193.2 Table 3: Motion-planning and navigation performance. All methods use identical maps, observations, action constraints, candidate sets, and planning budgets. We stress-test ELWM along two complementary held-out axes to test the learned transition reliability beyond the training configurations. The held-out-regime split changes the robot–environment interaction conditions, whereas the held-out-scene split changes the surrounding geometry. Both evaluations use the same 0.8 s recorded-action rollout horizon. A randomly initialized ELWM with the same architecture serves only as a sanity reference, so Table 2 measures whether training yields action-conditioned prediction across the two held-out splits. Scalability and Ablation Studies Scalability. We investigate whether the learning behavior of ELWM depends on tuned optimization setting. Specifically, ELWM and the Generic WM baseline are trained under a matched protocol while varying combinations of the learning rate, weight decay, and the SIGReg coefficient λsig _sig; all other data, optimization, and evaluation settings are kept fixed. As shown in Figure 5, ELWM consistently reaches a lower-loss region within the first few epochs and maintains lower training loss, validation loss, and future-latent consistency error across the tested configurations. The small variation between ELWM runs further indicates that this behavior is not tied to a single parameter combination. Meanwhile, the two model families eventually reach a comparable range of observable-motion error, showing that the improvement in latent prediction is not obtained by sacrificing motion-level accuracy. These results demonstrate that the physics-structured transition provides stable and parameter-robust learning across the evaluated optimization configurations. Figure 5: Convergence analysis across scalability training configurations. Compared in terms of (A) validation loss, (B) training loss, (C) latent-consistency error, and (D) observable-motion error, demonstrating ELWM stable learning beyond the hyperparameter setting. Ablations. Table 4 isolates where the PC-NTF gain comes from. The ANTF row measures geometry-only propagation. Replacing pH dynamics with a capacity-matched generic transition tests the ELWM structure; neutralizing the ELWM condition tests whether learned physics actually reaches the time field; and the scalar-only variant separates global speed calibration from spatial capability and candidate-specific traversal delay. Variant NRMSE↓ SPL↑ Phys. Coll.↓ Latency (ms)↓ ANTF 1.269 0.597 14.35% 40 PC-NTF w/o physical condition 2.347 0.483 18.75% 279 PC-NTF w/ shuffled MtM_t 1.036 0.580 22.11% 266 PC-NTF w/ generic WM 1.015 0.654 6.54% 479 ELWM + PC-NTF 1.012 0.661 5.79% 278 Table 4: Ablation study of PC-NTF. Prediction horizon and latency is measured per replanning cycle. Conclusion In this work, we investigated how interaction-derived physical knowledge can be represented by latent world models and translated into executable guidance for motion planning in unknown environments. We propose the Energy-Structured Latent World Model (ELWM) that structures action-conditioned prediction through port-Hamiltonian dynamics with explicit dissipation and control ports. For reusable physical regularities learning, we introduced Physics-Conditioned Neural Time Fields (PC-NTF) as the navigating motion system realization. PC-NTF connects predicted motion consequences to the global arrival-time representation of Neural Time Fields through a physically interpretable time-to-go objective. This construction preserves the efficiency and geometric guidance of neural time fields while allowing changes in robot–environment dynamics to inform motion selection. Our two-level evaluation analyses physically consistent world-model prediction and closed-loop motion planning, providing evaluation for ELWM physical consistent latent prediction quality and system-level planning benefit. The physical core evaluation also supports ELWM prediction and physical consistency on held-out scopes; paired navigation evaluation provides the corresponding benefit of PC-NTF motion planning. Future Work. We are delighted to extend PC-NTF from planar navigation to anisotropic, state-space, and contact-rich motion planning, and to evaluate transfer across broader robot configurations and real-world environments. References R. Balestriero and Y. LeCun (2025) Lejepa: provable and scalable self-supervised learning without the heuristics. arXiv preprint arXiv:2511.08544. Cited by: Training.. A. Bar, G. Zhou, D. Tran, T. Darrell, and Y. LeCun (2025) Navigation world models. In 2025 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), p. 15791–15801. 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