# Physics-Informed Neural Networks for Vessel Trajectory Prediction: Learning Time-Discretized Kinematic Dynamics via Finite Differences

arXiv: https://arxiv.org/abs/2506.12029 · HTML: https://mlsanity.com/paper/2506.12029/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 38 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 8 of 8)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [TrAISformer -- A Transformer Network with Sparse Augmented Data Representation and Cross Entropy Loss for AIS-based Vessel Trajectory Prediction](https://mlsanity.com/paper/2109.03958/) (2021)
- [Multi-Path Long-Term Vessel Trajectories Forecasting with Probabilistic Feature Fusion for Problem Shifting](https://mlsanity.com/paper/2310.18948/) (2023)
- [Enhancing Global Maritime Traffic Network Forecasting with Gravity-Inspired Deep Learning Models](https://mlsanity.com/paper/2401.13098/) (2024)
- [Maritime Tracking Data Analysis and Integration with AISdb](https://mlsanity.com/paper/2407.08082/) (2024)
- [Towards Real-World Validation of a Physics-Based Ship Motion Prediction Model](https://mlsanity.com/paper/2501.13804/) (2025)
- [Interpretable Data-Driven Ship Dynamics Model: Enhancing Physics-Based Motion Prediction with Parameter Optimization](https://mlsanity.com/paper/2502.18696/) (2025)

## Built on by (most cited first, 3 of 3)

- [Goal-Conditioned Reinforcement Learning for Data-Driven Maritime Navigation](https://mlsanity.com/paper/2509.01838/) (2025)
- [Multi-vessel Interaction-Aware Trajectory Prediction and Collision Risk Assessment](https://mlsanity.com/paper/2509.01836/) (2025)
- [A Retrieval-Enhanced Transformer for Multi-Step Port-of-Call Sequence Prediction in Global Liner Shipping](https://mlsanity.com/paper/2605.15937/) (2026)

## Abstract (arXiv)

Accurate vessel trajectory prediction is crucial for navigational safety, route optimization, traffic management, search and rescue operations, and autonomous navigation. Traditional data-driven models lack real-world physical constraints, leading to forecasts that disobey vessel motion dynamics, such as in scenarios with limited or noisy data where sudden course changes or speed variations occur due to external factors. To address this limitation, we propose a Physics-Informed Neural Network (PINN) approach for trajectory prediction that integrates a streamlined kinematic model for vessel motion into the neural network training process via a first- and second-order, finite difference physics-based loss function. This loss function, discretized using the first-order forward Euler method, Heun's second-order approximation, and refined with a midpoint approximation based on Taylor series expansion, enforces fidelity to fundamental physical principles by penalizing deviations from expected kinematic behavior. We evaluated PINN using real-world AIS datasets that cover diverse maritime conditions and compared it with state-of-the-art models. Our results demonstrate that the proposed method reduces average displacement errors by up to 32% across models and datasets while maintaining physical consistency. These results enhance model reliability and adherence to mission-critical maritime activities, where precision translates into better situational awareness in the oceans.