# PhysORD: A Neuro-Symbolic Approach for Physics-infused Motion Prediction in Off-road Driving

arXiv: https://arxiv.org/abs/2404.01596 · HTML: https://mlsanity.com/paper/2404.01596/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 29 references that are in the map.
- First posted on arXiv 2 Apr 2024.

## Builds on (most cited first, 19 of 19)

- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Dream to Control: Learning Behaviors by Latent Imagination](https://mlsanity.com/paper/1912.01603/) (2019)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [A Survey of Deep Learning Techniques for Autonomous Driving](https://mlsanity.com/paper/1910.07738/) (2019)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [TartanDrive: A Large-Scale Dataset for Learning Off-Road Dynamics Models](https://mlsanity.com/paper/2205.01791/) (2022)
- [Variational Integrator Networks for Physically Structured Embeddings](https://mlsanity.com/paper/1910.09349/) (2019)
- [PyPose: A Library for Robot Learning with Physics-based Optimization](https://mlsanity.com/paper/2209.15428/) (2022)
- [TartanDrive 2.0: More Modalities and Better Infrastructure to Further Self-Supervised Learning Research in Off-Road Driving Tasks](https://mlsanity.com/paper/2402.01913/) (2024)
- [OFFSEG: A Semantic Segmentation Framework For Off-Road Driving](https://mlsanity.com/paper/2103.12417/) (2021)
- [Probabilistic Traversability Model for Risk-Aware Motion Planning in Off-Road Environments](https://mlsanity.com/paper/2210.00153/) (2022)
- [PIAug -- Physics Informed Augmentation for Learning Vehicle Dynamics for Off-Road Navigation](https://mlsanity.com/paper/2311.00815/) (2023)
- [Lie Group Forced Variational Integrator Networks for Learning and Control of Robot Systems](https://mlsanity.com/paper/2211.16006/) (2022)
- [PyPose v0.6: The Imperative Programming Interface for Robotics](https://mlsanity.com/paper/2309.13035/) (2023)
- [Guaranteeing Consistency in a Motion Planning and Control Architecture Using a Kinematic Bicycle Model](https://mlsanity.com/paper/1804.08290/) (2018)
- [Multimodal dynamics modeling for off-road autonomous vehicles](https://mlsanity.com/paper/2011.11751/) (2020)
- [Forced Variational Integrator Networks for Prediction and Control of Mechanical Systems](https://mlsanity.com/paper/2106.02973/) (2021)
- [Double-Iterative Gaussian Process Regression for Modeling Error Compensation in Autonomous Racing](https://mlsanity.com/paper/2305.07740/) (2023)

## Built on by (most cited first, 7 of 7)

- [Imperative Learning: A Self-supervised Neuro-Symbolic Learning Framework for Robot Autonomy](https://mlsanity.com/paper/2406.16087/) (2024)
- [Four Principles for Physically Interpretable World Models](https://mlsanity.com/paper/2503.02143/) (2025)
- [PIETRA: Physics-Informed Evidential Learning for Traversing Out-of-Distribution Terrain](https://mlsanity.com/paper/2409.03005/) (2024)
- [Aligning Perception, Reasoning, Modeling and Interaction: A Survey on Physical AI](https://mlsanity.com/paper/2510.04978/) (2025)
- [FusionForce: End-to-end Differentiable Neural-Symbolic Layer for Trajectory Prediction](https://mlsanity.com/paper/2502.10156/) (2025)
- [Situationally-Aware Dynamics Learning](https://mlsanity.com/paper/2505.19574/) (2025)
- [An Alternative Trajectory for Generative AI](https://mlsanity.com/paper/2603.14147/) (2026)

## Abstract (arXiv)

Motion prediction is critical for autonomous off-road driving, however, it presents significantly more challenges than on-road driving because of the complex interaction between the vehicle and the terrain. Traditional physics-based approaches encounter difficulties in accurately modeling dynamic systems and external disturbance. In contrast, data-driven neural networks require extensive datasets and struggle with explicitly capturing the fundamental physical laws, which can easily lead to poor generalization. By merging the advantages of both methods, neuro-symbolic approaches present a promising direction. These methods embed physical laws into neural models, potentially significantly improving generalization capabilities. However, no prior works were evaluated in real-world settings for off-road driving. To bridge this gap, we present PhysORD, a neural-symbolic approach integrating the conservation law, i.e., the Euler-Lagrange equation, into data-driven neural models for motion prediction in off-road driving. Our experiments showed that PhysORD can accurately predict vehicle motion and tolerate external disturbance by modeling uncertainties. The learned dynamics model achieves 46.7% higher accuracy using only 3.1% of the parameters compared to data-driven methods, demonstrating the data efficiency and superior generalization ability of our neural-symbolic method.