  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. It outperforms existing methods both in accuracy and
efficiency and demonstrates data-efficient learning and generalization ability
in long-term prediction.
