Weakly-supervised Learning for Physics-informed Neural Motion Planning via Sparse Roadmap

Ruiqi Ni; Yuchen Liu; Ahmed H. Qureshi

cs.ROApr 14, 2026

Weakly-supervised Learning for Physics-informed Neural Motion Planning via Sparse Roadmap

Authors: Ruiqi Ni, Yuchen Liu, Ahmed H. Qureshi

Organizations: Department of Computer Science, Purdue University, West Lafayette, IN, USA, 47907.

Abstract

The motion planning problem requires finding a collision-free path between start and goal configurations in high-dimensional, cluttered spaces. Recent learning-based methods offer promising solutions, with self-supervised physics-informed approaches such as Neural Time Fields (NTFields) solving the Eikonal equation to learn value functions without expert demonstrations. However, existing physics-informed methods struggle to scale in complex, multi-room environments, where simply increasing the number of samples cannot resolve local minima or guarantee global consistency. We propose Hierarchical Neural Time Fields (H-NTFields), a weakly-supervised framework that combines weak supervision from sparse roadmaps with physics-informed PDE regularization. The roadmap provides global topological anchors through upper and lower bounds on travel times, while PDE losses enforce local geometric fidelity and obstacle-aware propagation. Experiments on 18 Gibson environments and real robotic platforms show that H-NTFields substantially improves robustness over prior physics-informed methods, while enabling fast amortized inference through a continuous value representation.

Figures & tables

Panel of Figure 1 — Fig. 1 : Comparison of speed and time fields in a Gibson environment. The full model produces smooth time fields with correct obstacle-aware connectivity, closely matching the ground-truth from FMM. In contrast, the PDE-only variant collapses into local minima (lower-left region), while the roadmap-only variant preserves global contours but fails to capture fine obstacle geometry.

Methods	Performance Metrics
Methods	Time (sec) $\downarrow$	Length	SR (%) $\uparrow$
Ours	$0.115\pm 0.071$	$0.72\pm 0.46$	90.8
Roadmap only	$0.105\pm 0.071$	$0.77\pm 0.52$	68.5
PDE only	$0.089\pm 0.053$	$0.68\pm 0.46$	60.0

TABLE I : Quantitative ablation study on Gibson environments. The full H-NTFields model achieves the highest success rate while maintaining comparable planning time to the ablated variants. PDE-only reduces average path length slightly but fails in long-horizon tasks, while roadmap-only improves global consistency yet misses fine obstacle fidelity, leading to lower overall performance.

Data	Performance Metrics
Data	Sample $\downarrow$	Roadmap $\downarrow$	SR (Ours) $\uparrow$	SR (PDE) $\uparrow$
10k	0.23	14.3	61.4	20.0
20k	0.26	16.3	82.5	26.6
50k	0.32	17.6	90.8	59.8
100k	0.42	18.1	92.7	75.1
200k	0.63	18.4	92.0	76.8

TABLE II : Scalability study on Gibson environments. H-NTFields achieves 90.8% success with only 50k training pairs, while TD-NTFields requires 200k samples to reach just 76.8%—lower than H-NTFields trained with only 20k. Reported costs (seconds) include sample generation and roadmap construction/query, showing that combining sparse roadmap supervision with PDE regularization yields both efficiency and scalability.

Fig. 2 : Path comparison in a Gibson environment. The figure shows paths generated by different planners for the same start–goal query. NTFields, P-NTFields, and TD-NTFields fail to produce viable paths. Our method successfully generates a smooth trajectory (orange) in 0.22s. RRTConnect (yellow) and Lazy-PRM (cyan) find valid paths but are slower (3.33s and 3.99s). FMM (green) runs faster (0.98s) but produces paths with some sharp turns due to grid discretization.

Methods	Performance Metrics
Methods	Time (sec) $\downarrow$	Length	SR (%) $\uparrow$
Ours	$0.115\pm 0.071$	$0.72\pm 0.46$	90.8
NTFields	$0.076\pm 0.053$	$0.75\pm 0.94$	27.2
P-NTFields	$0.076\pm 0.064$	$0.74\pm 0.85$	37.2
TD-NTFields	$0.081\pm 0.051$	$0.72\pm 0.46$	76.8
RRTConnect	$1.165\pm 1.876$	$0.71\pm 0.41$	98.8

TABLE III : Comparison for all motion planning methods in 18 Gibson environments. It can be seen that our method exhibits much higher SR than other physics-informed methods with relatively slower planning times. The relatively slow planning times are due to our method solving much harder cases, where the start and goal span across multiple rooms.

Physics-Informed ML

Physics-informed machine learning combines data-driven models with physical laws to solve, predict, and infer scientific systems.

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Energy-Structured Latent World Models with Neural Time Fields for Physically Constistent Open-World Motion Planning

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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.

Abstract

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