Accurate prediction of off-road vehicle motion over deformable terrain remains challenging because sinkage, slip, and traction vary with local soil conditions. Existing learning-based kinodynamic models directly approximate vehicle-terrain interactions from data but do not explicitly represent soil mechanics and offer limited physical interpretability. To address these limitations, we present NeSAM, a neuro-symbolic framework that combines differentiable Bekker-Wong terramechanics with learned terrain representations and a Transformer-based residual dynamics model for long-horizon, six degree-of-freedom kinodynamic prediction. The terramechanics component models soil-dependent interaction forces, while the residual model corrects discrepancies between the analytical prediction and the observed vehicle dynamics. NeSAM further estimates physically meaningful soil parameters from terrain observations and updates them online using an extended Kalman filter. We evaluate NeSAM in Verti-Bench, a simulator built on the Chrono multiphysics engine, and validate its performance on a physical Verti-4-Wheeler platform. NeSAM improves prediction accuracy by up to 30% in simulation and 29% on real-world data relative to the strongest compared baselines. When integrated with a close-loop navigation controller, NeSAM further improves traversal success rate through online soil adaptation while reduces Hausdorff distance to the reference trajectory by 69.4%, indicating improved trajectory tracking accuracy.

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