# Physics-informed Temporal Difference Metric Learning for Robot Motion Planning

arXiv: https://arxiv.org/abs/2505.05691 · HTML: https://mlsanity.com/paper/2505.05691/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 61 references that are in the map.
- First posted on arXiv 9 May 2025.

## Builds on (most cited first, 30 of 40)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Point Transformer](https://mlsanity.com/paper/2011.00931/) (2020)
- [Learning Invariant Representations for Reinforcement Learning without Reconstruction](https://mlsanity.com/paper/2006.10742/) (2020)
- [PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies](https://mlsanity.com/paper/2206.04670/) (2022)
- [DD-PPO: Learning Near-Perfect PointGoal Navigators from 2.5 Billion Frames](https://mlsanity.com/paper/1911.00357/) (2019)
- [Leveraging Demonstrations for Deep Reinforcement Learning on Robotics Problems with Sparse Rewards](https://mlsanity.com/paper/1707.08817/) (2017)
- [iGibson 2.0: Object-Centric Simulation for Robot Learning of Everyday Household Tasks](https://mlsanity.com/paper/2108.03272/) (2021)
- [Optimal Goal-Reaching Reinforcement Learning via Quasimetric Learning](https://mlsanity.com/paper/2304.01203/) (2023)
- [Neural Kinematic Networks for Unsupervised Motion Retargetting](https://mlsanity.com/paper/1804.05653/) (2018)
- [Learning Sampling Distributions for Robot Motion Planning](https://mlsanity.com/paper/1709.05448/) (2017)
- [Motion Planning Networks](https://mlsanity.com/paper/1806.05767/) (2018)
- [PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks](https://mlsanity.com/paper/2402.00326/) (2024)
- [Motion Planning Networks: Bridging the Gap Between Learning-based and Classical Motion Planners](https://mlsanity.com/paper/1907.06013/) (2019)
- [Motion Policy Networks](https://mlsanity.com/paper/2210.12209/) (2022)
- [A Geometric Perspective on Optimal Representations for Reinforcement Learning](https://mlsanity.com/paper/1901.11530/) (2019)
- [Path Planning using Neural A* Search](https://mlsanity.com/paper/2009.07476/) (2020)
- [EDMP: Ensemble-of-costs-guided Diffusion for Motion Planning](https://mlsanity.com/paper/2309.11414/) (2023)
- [Robot Motion Planning in Learned Latent Spaces](https://mlsanity.com/paper/1807.10366/) (2018)
- [Batch Informed Trees (BIT*): Sampling-based Optimal Planning via the Heuristically Guided Search of Implicit Random Geometric Graphs](https://mlsanity.com/paper/1405.5848/) (2014)
- [iPlanner: Imperative Path Planning](https://mlsanity.com/paper/2302.11434/) (2023)
- [Model-Predictive Control via Cross-Entropy and Gradient-Based Optimization](https://mlsanity.com/paper/2004.08763/) (2020)
- [Fast Marching Tree: a Fast Marching Sampling-Based Method for Optimal Motion Planning in Many Dimensions](https://mlsanity.com/paper/1306.3532/) (2013)
- [Differentiable Spatial Planning using Transformers](https://mlsanity.com/paper/2112.01010/) (2021)
- [An Inductive Bias for Distances: Neural Nets that Respect the Triangle Inequality](https://mlsanity.com/paper/2002.05825/) (2020)
- [Improved Representation of Asymmetrical Distances with Interval Quasimetric Embeddings](https://mlsanity.com/paper/2211.15120/) (2022)
- [Neural Path Planning: Fixed Time, Near-Optimal Path Generation via Oracle Imitation](https://mlsanity.com/paper/1904.11102/) (2019)
- [NTFields: Neural Time Fields for Physics-Informed Robot Motion Planning](https://mlsanity.com/paper/2210.00120/) (2022)
- [Attention Beats Concatenation for Conditioning Neural Fields](https://mlsanity.com/paper/2209.10684/) (2022)
- [Deeply Informed Neural Sampling for Robot Motion Planning](https://mlsanity.com/paper/1809.10252/) (2018)

## Built on by (most cited first, 6 of 6)

- [A Survey of Agentic Reasoning for Large Language Models: Towards Recursively Self-Improving and Collective Agents](https://mlsanity.com/paper/2601.12538/) (2026)
- [Physics-informed Value Learner for Offline Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/2509.06782/) (2025)
- [Mollified Value Learning](https://mlsanity.com/paper/2602.23280/) (2026)
- [Toward Generalist Neural Motion Planners for Robotic Manipulators: Challenges and Opportunities](https://mlsanity.com/paper/2603.24318/) (2026)
- [Neural Navigation Functions for Zero-Shot Generalizable Motion Planning](https://mlsanity.com/paper/2606.03756/) (2026)
- [PoseShield: Neural Collision Fields for Human Self-Collision Resolution](https://mlsanity.com/paper/2606.29686/) (2026)

## Abstract (arXiv)

The motion planning problem involves finding a collision-free path from a robot's starting to its target configuration. Recently, self-supervised learning methods have emerged to tackle motion planning problems without requiring expensive expert demonstrations. They solve the Eikonal equation for training neural networks and lead to efficient solutions. However, these methods struggle in complex environments because they fail to maintain key properties of the Eikonal equation, such as optimal value functions and geodesic distances. To overcome these limitations, we propose a novel self-supervised temporal difference metric learning approach that solves the Eikonal equation more accurately and enhances performance in solving complex and unseen planning tasks. Our method enforces Bellman's principle of optimality over finite regions, using temporal difference learning to avoid spurious local minima while incorporating metric learning to preserve the Eikonal equation's essential geodesic properties. We demonstrate that our approach significantly outperforms existing self-supervised learning methods in handling complex environments and generalizing to unseen environments, with robot configurations ranging from 2 to 12 degrees of freedom (DOF).