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

AI summary

Overview

Research area: Applied machine learning for maritime engineering — specifically, physics-informed/hybrid modeling for main engine power prediction of ships.

Technical level: Intermediate. The machine learning (XGBoost, neural networks, PINNs, residual learning) is explained with equations but the underlying idea is accessible; some familiarity with regression and automatic differentiation helps.

Scope (one sentence): The paper introduces and evaluates a hybrid framework that combines sea-trial-derived calm-water power curves with learned residual corrections, comparing XGBoost, a simple Neural Network, and a Physics-Informed Neural Network (PINN) in both hybrid and pure data-driven (baseline) form on roughly 40,000 points of in-service ship data.

What This Paper Is About

Accurate prediction of a vessel's main engine power matters for fuel efficiency, emissions compliance and performance optimization, but purely data-driven models tend to be unreliable when asked to predict outside the operating conditions seen during training — they can produce power–speed curves that violate the fundamental propeller law (the cubic-ish power–speed relationship). This paper addresses that extrapolation weakness by splitting the prediction task in two: a physics-based baseline taken from sea trials captures the dominant power–speed dependence, and a machine learning model learns only the residual (the deviation caused by wind, waves, trim, draft, fouling and ageing).

Key Contributions

1. A hybrid modeling framework for vessel main engine power prediction that uses sea-trial calm-water power curves as a physically consistent baseline and trains a nonlinear regressor to predict only the residual correction, improving extrapolation stability and physical consistency relative to purely data-driven approaches.
2. A comparative study applying this hybrid framework to three different model paradigms — XGBoost, a simple Neural Network, and a Physics-Informed Neural Network (PINN) — alongside identical models without the baseline component, demonstrating the framework's versatility across modeling families.
3. A formulation of the PINN physics loss for the hybrid setting, where the derivative constraint on the network output is expressed relative to the sea-trial baseline (Eqs. 9–11), with all derivatives computed using PyTorch's automatic differentiation engine.
4. Validation on in-service data from real vessel operations (approximately 40,000 data points from five months of operation), positioning the framework as a computationally efficient tool for maritime optimization such as weather routing and fuel efficiency planning.

Main Findings

• Quantitative differences are small, and slightly favor the baseline models on global error metrics. Across all architectures the baseline (non-hybrid) variants achieved marginally lower training errors — for example, XGBoost baseline Train MAE 114.2 kW versus hybrid 143.3 kW, and Test MAE 122.2 kW versus 148.8 kW. The authors describe these differences as small and within acceptable limits for operational maritime applications, and note in the conclusion that they are well under 1% of the vessel's maximum power.
• Reported error magnitudes vary by architecture. On the test set, XGBoost baseline recorded MAE 122.2 kW and RMSE 195.0 kW; NN baseline recorded MAE 162.66 kW and RMSE 225.10 kW; PINN baseline recorded MAE 144.30 kW and RMSE 211.89 kW. The corresponding hybrid figures were XGBoost MAE 148.8 kW / RMSE 208.2 kW, NN MAE 219.32 kW / RMSE 284.33 kW, and PINN MAE 171.19 kW / RMSE 229.45 kW.
• The real advantage of hybridization is extrapolation behavior, not error metrics. Evaluated across speeds of 8–17 kn at ballast draft with 5-kn wind and wind directions of 0°, 90° and 180°, baseline models flattened, became non-monotonic, or showed wind-direction sensitivity in sparsely sampled regions, violating the expected propeller-law scaling.
• Hybrid XGBoost removes non-physical flattening. The baseline XGBoost model frequently showed non-monotonic or flattened power–speed behavior at higher speeds where training data were unavailable; anchoring to the calm-water baseline restored a smooth, monotonic increase in power.
• Hybrid NN reduces high-speed overestimation and wind-direction artifacts. Because it learns only residual deviations around a physically consistent baseline, the resulting curves were smoother and generalized better across the full speed range.
• Hybrid PINN performed best overall. It enforces consistency with both the sea-trial power law and the PINN derivative constraints, producing physically coherent predictions across all wind directions throughout the 8–17 kn range. The authors identify it as the best balance between numerical accuracy and physical consistency.
• Framing of the benefit. The paper states the main benefit of the hybrid formulations is "regularization of model behavior in sparsely sampled and extrapolative regimes" rather than a reduction in error metrics.

Methodology in Plain English

• Build a physics baseline from sea trials. During sea trials, a ship's power and speed are measured at specific draft conditions, typically in ballast (T_b) and laden (T_l) states. The relationship is approximated as a power law P = cV^n, where V is speed through water and c and n are vessel-specific coefficients. The coefficients are obtained by taking the natural logarithm of both sides, ln(P) = ln(c) + n·ln(V), and performing a linear regression on the log-transformed sea trial data (Appendix A).
• Interpolate for intermediate drafts. For drafts between ballast and laden, the baseline power is computed by linear interpolation between the two sea-trial curves using Eq. (3), weighted by how far the actual draft T lies between T_b and T_l. The authors acknowledge this linear interpolation is a simplification of the underlying physics but call it a robust baseline that satisfies the ballast and laden conditions simultaneously.
• Train a model on the residual only. The final prediction is the sum of the physics baseline and a learned correction term f(X) (Eq. 4). The residual is the difference between measured main engine power and the sea-trial baseline power. Input features include parameters not captured by sea trials, such as wind speed and direction, wave height, vessel ageing, and time since last dry-docking.
• Make the inputs wind-friendly. Wind true speed and wind true direction were decomposed into W_x = WTS·cos(WTD) and W_y = WTS·sin(WTD) components, following the earlier work of Bourchas and Papalambrou (2025).
• PINN loss design. The total PINN loss is L_PINN = L_data + λ·L_P.law (Eq. 6), where L_data is mean squared prediction error, λ is a weighting coefficient fixed to 100, and the physics loss penalizes deviation of the network's speed derivative from the expected propeller-law derivative, 3cV² (Eq. 8), adapted to the hybrid decomposition.
• Train everything the same way for fairness. Both baseline and hybrid versions of all three architectures went through a unified hyperparameter optimization procedure. XGBoost used Scikit-learn's RandomizedSearchCV with negative mean squared error as the scoring metric, exploring learning rates from 0.01 to 0.30, max depth 3–10, n_estimators 5–500, and L1/L2 regularization in {0, 1, 100}; inputs and outputs were standardized with StandardScaler fitted on the training set. NN and PINN models were built in PyTorch, tuned via Weights & Biases sweep functionality with Bayesian optimization minimizing test-set RMSE, exploring learning rates of 1×10⁻², 1×10⁻³, 3×10⁻⁴ and 1×10⁻⁴, layer widths of 64, 128 or 256 neurons, and depths of 4, 6 or 8 layers. Each model was trained with the Adam optimizer for 1000 epochs, without learning rate scheduling or early stopping.
• Data split and hardware. The approximately 40,000-point dataset was randomly split 80%-10%-10% into training, validation and test sets. Experiments ran on Windows 10 Pro with an AMD Ryzen 5500 CPU, an NVIDIA RTX 3060 GPU with 12 GB VRAM, and 32 GB of RAM.

Why This Matters

Impact on research. The paper contributes to the physics-informed machine learning literature by demonstrating a residual-hybrid strategy as a lower-friction alternative to fully physics-constrained models. The authors note that PINNs, while capable of enforcing physical consistency, are extremely hard to train and can struggle to capture the steep gradients and discontinuities present in noisy real-world operational data in the forward problem. Relieving the ML component of the burden of rediscovering the dominant power–speed law makes the learning task simpler and better regularized.

Real-world applications (as identified in the paper):

• Weather routing, where voyage decisions are made precisely in regions with limited historical data coverage.
• Trim optimization.
• Energy efficiency planning.
• Vessel performance monitoring, including support for compliance with emission regulations such as IMO 2020.

Industry relevance. Maritime transportation is described as the backbone of international trade, and its carbon footprint has led to increasingly stringent regulatory frameworks. The framework offers a practical, computationally efficient tool for decision support where accurate power prediction is required even when no historical data are available — something global error metrics alone do not guarantee.

Future Directions

• Tune the physics-loss weight. The weighting coefficient λ was fixed at 100 in this work for simplicity and stability; the authors explicitly state it can be treated as a tunable hyperparameter in future studies.
• Refine the draft interpolation. The linear interpolation between ballast and laden drafts is acknowledged as a simplification of the underlying physics, leaving room for more physically faithful intermediate-draft baselines.
• Extend beyond global error metrics. Since the reported MAE and RMSE differences between baseline and hybrid models are modest and sometimes favor the baselines, the paper implicitly raises the question of what metric best captures extrapolation reliability, which its qualitative analysis addresses but does not quantify.
• Broaden the evidence base. The study relies on a single roughly 40,000-point dataset covering five months of operation from the earlier work of Bourchas and Papalambrou (2025); generalization to other vessels, longer time spans, and additional operational conditions is not established here.

Target Audience

Maritime and naval engineering researchers working on vessel performance and energy efficiency; machine learning researchers interested in physics-informed and hybrid residual modeling; data scientists in shipping companies and fleet operators building performance-monitoring or weather-routing decision-support tools; and graduate students in marine engineering or applied ML who want a concrete case study of embedding domain knowledge into a regression pipeline. Readers focused purely on squeezing out the lowest possible error metric may find the paper's message inverted: its distinctive claim is that physical consistency under extrapolation matters more than marginal gains in MAE and RMSE.

Authors’ abstract

Accurate prediction of main engine power is essential for vessel performance optimization, fuel efficiency, and compliance with emission regulations. Conventional machine learning approaches, such as Support Vector Machines, variants of Artificial Neural Networks (ANNs), and tree-based methods like Random Forests, Extra Tree Regressors, and XGBoost, can capture nonlinearities but often struggle to respect the fundamental propeller law relationship between power and speed, resulting in poor extrapolation outside the training envelope. This study introduces a hybrid modeling framework that integrates physics-based knowledge from sea trials with data-driven residual learning. The baseline component, derived from calm-water power curves of the form $P = cV^n$, captures the dominant power-speed dependence, while another, nonlinear, regressor is then trained to predict the residual power, representing deviations caused by environmental and operational conditions. By constraining the machine learning task to residual corrections, the hybrid model simplifies learning, improves generalization, and ensures consistency with the underlying physics. In this study, an XGBoost, a simple Neural Network, and a Physics-Informed Neural Network (PINN) coupled with the baseline component were compared to identical models without the baseline component. Validation on in-service data demonstrates that the hybrid model consistently outperformed a pure data-driven baseline in sparse data regions while maintaining similar performance in populated ones. The proposed framework provides a practical and computationally efficient tool for vessel performance monitoring, with applications in weather routing, trim optimization, and energy efficiency planning.