Scientific Knowledge-Guided Machine Learning for Vessel Power Prediction: A Comparative Study

Conference: AAAI 2026 arXiv: 2602.18403 Authors: Orfeas Bourchas, George Papalambrou Code: None Area: Scientific ML / Marine Engineering Keywords: vessel power prediction, hybrid modeling, physical residual learning, PINN, XGBoost, extrapolation generalization

TL;DR

A hybrid modeling framework combining a physics baseline with a data-driven residual is proposed. The sea trial power curve (propeller law \(P=cV^n\)) serves as the baseline, and XGBoost/NN/PINN models learn the residual correction, significantly improving extrapolation stability and physical consistency in sparse data regions.

Background & Motivation

1. Tightening emission regulations: IMO constraints on shipping carbon emissions are increasingly stringent, making accurate main engine power prediction a prerequisite for fuel optimization and compliance.
2. Limitations of traditional empirical models: Sea trial curves and calm-water resistance models capture only the cubic speed–power relationship, failing to reflect complex operational deviations such as wind, waves, and hull fouling.
3. Poor extrapolation of pure data-driven models: Models such as XGBoost and ANN achieve high accuracy within the training distribution, but predictions in sparse regions—such as high-speed or extreme operating conditions—often violate the propeller law, exhibiting non-monotonic or flat curves.
4. Training difficulties of PINNs: Although physics-informed neural networks can incorporate physical constraints, training on noisy measured data is extremely challenging, and capturing steep gradients and discontinuities remains difficult.
5. Insights from residual hybrid strategies: Recent work in physics-informed learning, such as Stacked Residual approaches, demonstrates that decomposing complex predictions into a physics baseline plus residual correction substantially simplifies the learning task.
6. Lack of systematic cross-architecture comparison: Prior work has not compared XGBoost, NN, and PINN models under a unified framework before and after the introduction of a physics baseline.

Mechanism

Goal:

Method

Overall Architecture

The power prediction is decomposed into two components:

\[\hat{P}(\bar{X}) = P_{\text{sea trial}}(V, T) + f(\bar{X})\]

• Physics baseline \(P_{\text{sea trial}}(V, T)\): A calm-water power curve obtained from sea trials, of the form \(P = cV^n\), fitted separately for ballast and laden conditions, with intermediate drafts computed by linear interpolation:

\[P_{\text{sea trial}}(V,T) = \left(1-\frac{T-T_b}{T_l-T_b}\right) P_b(V) + \frac{T-T_b}{T_l-T_b} P_l(V)\]

• Residual regressor \(f(\bar{X})\): A data-driven model that learns only the difference (residual) between the measured power and the physics baseline, with inputs including wind velocity components \(W_x, W_y\), hull aging time, and other factors not covered by the baseline.

Hybrid Configurations for Three Regressors

1. Hybrid XGBoost: XGBoost predicts the residual; hyperparameters are optimized via RandomizedSearchCV (learning rate, tree depth, number of estimators, L1/L2 regularization).
2. Hybrid NN: A simple fully connected network predicts the residual, trained with Adam for 1000 epochs; hyperparameters are optimized via WandB Bayesian sweep (learning rate, number of layers, number of neurons).
3. Hybrid PINN: A physical loss term is added on top of the NN. The total loss is:

\[\mathcal{L}_{\text{PINN}} = \mathcal{L}_{\text{data}} + \lambda \mathcal{L}_{P.\text{law}}\]

where the physical loss enforces the propeller law constraint via partial differentiation with respect to speed:

\[\mathcal{L}_{P.\text{law}} = \sum_{i}\left(\frac{\partial f}{\partial V}\bigg|_{x_i} - \left[3cV_i^2 - \text{baseline derivative term}\right]\right)^2\]

\(\lambda=100\) is a fixed weight; all derivatives are computed via PyTorch autograd.

Input Features

Symbol	Description	Unit
\(V\)	Speed through water (S.T.W.)	kn
\(T\)	Mean draft	m
Trim	Trim	m
\(W_x, W_y\)	cos/sin components of true wind speed	kn
\(t\)	Logging time (UTC)	—

Experiments

Dataset

Approximately 40,000 data points from five months of operational data from a real vessel, split 80/10/10 into training/validation/test sets.

Quantitative Results

Main Results

Metric	XGBoost Baseline	XGBoost Hybrid	NN Baseline	NN Hybrid	PINN Baseline	PINN Hybrid
Train MAE [kW]	114.2	143.3	181.0	235.3	147.0	214.7
Train RMSE [kW]	180.5	199.9	229.7	292.1	205.5	248.3
Val MAE [kW]	114.2	143.3	156.3	214.1	135.5	167.2
Val RMSE [kW]	180.5	199.9	212.9	272.4	204.2	219.5
Test MAE [kW]	122.2	148.8	162.7	219.3	144.3	171.2
Test RMSE [kW]	195.0	208.2	225.1	284.3	211.9	229.5

Hybrid models exhibit slightly higher global errors than baseline models (within 1% of the vessel's maximum power), though global metrics fail to capture extrapolation behavior in sparse regions.

Hyperparameter Comparison

Ablation Study

Hyperparameter	XGBoost Base	XGBoost Hybrid	NN Base	NN Hybrid	PINN Base	PINN Hybrid
Learning rate	0.25	0.1	1e-4	3e-4	1e-4	3e-4
Layers/depth	depth=10	depth=10	6 layers	4 layers	8 layers	6 layers
Neurons/estimators	40	500	256	64	256	256
Regularization	L1=1, L2=0	L1=1, L2=100	—	—	λ=100	λ=100

Qualitative Extrapolation Analysis

Extrapolation behavior is evaluated across the speed range of 8–17 kn under ballast draft and 5 kn wind conditions at wind directions of 0°/90°/180°: - XGBoost Baseline: Non-monotonic or flat power–speed curves frequently appear in high-speed regions, violating physical expectations. - NN Baseline: Overestimation in high-speed regions and anomalous sensitivity to wind direction. - PINN Baseline: Benefits from physical loss, yet still exhibits deviations in sparse regions. - All Hybrid models: Power–speed curves are smooth and monotonically increasing, maintaining physical consistency beyond the training envelope.

Key Findings

• The core value of the hybrid framework lies not in reducing global error metrics, but in regularizing model behavior in sparse and extrapolation regions.
• Hybrid PINN achieves the best performance across all architectures, combining competitive accuracy with the strongest physical consistency.
• Restricting the ML task to residual prediction is equivalent to a strong regularization: the model need not rediscover the dominant speed–power relationship.
• In practical applications such as weather routing and speed optimization, decisions are often made in regions with the sparsest historical data coverage—precisely where the hybrid model's advantage is greatest.

Key Findings

• The primary components/modules contribute the most critical performance improvements.

Highlights & Insights

• Concise and effective decomposition: Using the sea trial curve as a physics baseline and learning the residual with ML is intuitive and highly practical for engineering deployment.
• Unified cross-architecture validation: Effectiveness is demonstrated across three fundamentally different architectures—XGBoost, NN, and PINN—establishing the generality of the framework.
• Derivation of PINN physical loss: The physical constraint for the PINN (partial derivative of the residual function with respect to speed) is re-derived within the hybrid framework, with complete and rigorous mathematical derivation.
• Focus on extrapolation over fitting: Rather than pursuing lower training error, the paper targets physical credibility of model predictions under unseen operating conditions, reflecting a precise problem formulation.

Limitations & Future Work

• The physics baseline uses only linear interpolation between two draft conditions (ballast and laden), which is a simplification of the actual nonlinear draft–power relationship.
• \(\lambda=100\) is fixed throughout; the effect of adaptive weight scheduling on PINN training is not explored.
• Experimental data come from a single vessel over five months; generalization to different vessel types and longer time horizons remains to be verified.
• The cumulative effect of hull fouling over time is not explicitly modeled (it appears only as an input feature).
• The hybrid model depends on the availability of sea trial data, which is not standardized across all vessels.
• No comparison is made with more advanced time-series models such as Transformer or Mamba.

Related Work & Insights

• Classical ML prediction: Gaussian Process, SVR (Bassam et al. 2023; Gkerekos et al. 2019), ANN/LSTM (Guo et al. 2023; Cai et al. 2024; Chen et al. 2024), XGBoost (Nguyen et al. 2023; Agand et al. 2023; Fan et al. 2024).
• PINNs in the maritime domain: Raissi et al. (2017) introduced the PINN framework; Lang et al. (2024) and Bourchas & Papalambrou (2025) applied it to vessel speed/power prediction.
• Residual hybrid strategies: Howard et al. (2023) proposed Stacked Networks to improve PINN training; Eshkofti & Barreau (2025) proposed Vanishing Stacked-Residual PINN.

Rating

• Novelty: ⭐⭐⭐ The framework is intuitive; the core contribution lies in systematic comparison rather than methodological innovation.
• Experimental Thoroughness: ⭐⭐⭐⭐ Comprehensive comparison across three architectures × two configurations, including both quantitative and qualitative analysis.
• Writing Quality: ⭐⭐⭐⭐ Clear problem motivation, complete mathematical derivation, and rich figures and tables.
• Value: ⭐⭐⭐⭐ Provides a practical tool for vessel power prediction with direct applicability to weather routing and energy efficiency planning.

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