Function-Space Priors for Bayesian Neural ODEs with Application to Vessel Trajectory Prediction
Authors: Jaeyeong Lee, Wonmo Koo, Heeyoung Kim
Organizations: aDepartment of Industrial and Systems Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea.
Abstract
Vessel trajectory prediction from Automatic Identification System (AIS) data is essential for maritime situational awareness, yet it remains challenging due to irregular sampling, missing reports, and complex dynamics. Beyond accurate point forecasts, maritime applications also demand well-calibrated uncertainty estimates for reliable decision-making. Bayesian Neural Ordinary Differential Equations (ODEs) offer a principled framework for continuous-time trajectory modeling with uncertainty quantification by placing a prior over the neural vector field parameters. However, the commonly used isotropic Gaussian weight prior fails to encode informative structural properties of vessel dynamics, such as smoothness and locality. Existing function-space Bayesian neural network methods address this limitation for static mappings, but do not transfer directly to Neural ODEs, where the primary quantity of interest is the trajectory rather than the vector field itself. In principle, one could place a Gaussian process (GP) prior directly over ODE solutions, but this requires propagating distributions through a nonlinear ODE solver, which is analytically intractable. To address this challenge, we adopt a practical approach that imposes a GP-kernel-based prior directly on the vector field evaluated at a finite set of measurement points. Specifically, we augment the standard weight-space variational objective with a kernel-based regularizer that penalizes deviations of the vector field from the structure implied by a GP prior. To handle long and irregular AIS trajectories, we further combine this function-space regularization with probabilistic multiple shooting, which decouples inference across temporal segments while maintaining global consistency.
Accurate vessel trajectory prediction is essential for safe and efficient maritime operations, enabling collision avoidance and supporting route optimization. Although memory-augmented neural networks have recently shown strong performance in pedestrian and road-vehicle trajectory prediction by selectively retrieving relevant information from an external memory, their potential for vessel trajectory prediction remains underexplored. This paper presents an empirical investigation of memory-based trajectory prediction using Automatic Identification System (AIS) data. Experiments on data from the Gulf of Mexico and the New York Bight demonstrate consistent and substantial performance gains over a range of deep learning baselines that do not incorporate an external memory.
Accurate vessel trajectory prediction is critical for maritime safety and anomaly detection, yet existing models often struggle with geographic bias and navigational realism. We propose the Continuous Regression Hybrid Transformer (CRHT), a deep learning framework designed to forecast vessel motion using Automatic Identification System (AIS) data. To mitigate spatial data imbalance, we introduce an online K-means cluster sampling strategy that ensures diverse exposure to rare maneuvers during training. Our hybrid architecture integrates 1D convolutional layers for local kinematic feature extraction with a multi-head attention mechanism for global temporal context. CRHT demonstrates superior performance in short-term forecasting, achieving the lowest errors at the 1-hour horizon. The results demonstrate that while discrete models provide high navigational stability over long horizons, CRHT offers an optimal balance of precision and maneuver tracking for real-time maritime surveillance.
Alexander Schiøtz, Bertram Hage, Christian Rand +2
Spatiotemporal forecasting in physical systems, such as large-scale traffic networks, requires modeling a dual dynamic: continuous macroscopic rhythms and discrete, unpredictable microscopic shocks. While Neural Ordinary Differential Equations (ODEs) excel at capturing smooth evolution, their inherent Lipschitz continuity constraints inevitably cause severe over-smoothing when confronting abrupt anomalies. Recent physics-informed methods attempt to bypass this by penalizing numerical integration errors to enforce manifold smoothness. However, we mathematically reveal that such rigid regularization inherently triggers gradient conflicts and ``attention collapse,'' stripping the model of its sensitivity to anomalies. To resolve this continuity-shock dilemma, we propose Local Truncation Error-Guided Neural ODEs (LTE-ODE). Rather than treating numerical error as a nuisance to be eliminated, we innovatively repurpose the Local Truncation Error (LTE) as an unsupervised forward inductive bias. By mapping the LTE into a dynamic spatial attention mask, our architecture gracefully preserves high-precision continuous ODE evolution in stable regions, while adaptively triggering a discrete compensation branch exclusively at shock points. Trained purely end-to-end without manifold penalties, LTE-ODE achieves state-of-the-art performance on multiple large-scale benchmarks, exhibiting exceptional robustness against highly non-linear fluctuations. Furthermore, our ablation on integration steps demonstrates high deployment flexibility, allowing the model to seamlessly adapt to varying hardware memory constraints in real-world applications.