Physics-Informed Machine Learning for Vessel Shaft Power and Fuel Consumption Prediction: Interpretable KAN-based Approach

Hamza Haruna Mohammed; Dusica Marijan; Arnbjørn Maressa

cs.LGFeb 25, 2026

Physics-Informed Machine Learning for Vessel Shaft Power and Fuel Consumption Prediction: Interpretable KAN-based Approach

Authors: Hamza Haruna Mohammed, Dusica Marijan, Arnbjørn Maressa

Organizations: Simula Research Laboratory, Oslo, Norway · Navtor AS, Egersund, Norway

Abstract

Accurate prediction of shaft rotational speed, shaft power, and fuel consumption is crucial for enhancing operational efficiency and sustainability in maritime transportation. Conventional physics-based models provide interpretability but struggle with real-world variability, while purely data-driven approaches achieve accuracy at the expense of physical plausibility. This paper introduces a Physics-Informed Kolmogorov-Arnold Network (PI-KAN), a hybrid method that integrates interpretable univariate feature transformations with a physics-informed loss function and a leakage-free chained prediction pipeline. Using operational and environmental data from five cargo vessels, PI-KAN consistently outperforms the traditional polynomial method and neural network baselines. The model achieves the lowest mean absolute error (MAE) and root mean squared error (RMSE), and the highest coefficient of determination (R^2) for shaft power and fuel consumption across all vessels, while maintaining physically consistent behavior. Interpretability analysis reveals rediscovery of domain-consistent dependencies, such as cubic-like speed-power relationships and cosine-like wave and wind effects. These results demonstrate that PI-KAN achieves both predictive accuracy and interpretability, offering a robust tool for vessel performance monitoring and decision support in operational settings.

Figures & tables

Fig. 2 : Leakage-free chained prediction pipeline used in PI-KAN: shaft RPM is predicted first, followed by shaft power and then fuel consumption, with each stage receiving out-of-fold (OOF) predictions from the previous one to avoid target leakage and preserve physical causality.

Category	Feature	Description
Operational
	$V$	Speed through water (knots)
	$T$	Vessel draught (m), distance from waterline to keel
Environmental
	$depth_{\text{sea}}$	Sea depth below keel (m)
	$t_{\text{sea}}$	Sea surface temperature (°C)

TABLE I : Categorized input features used for shaft rpm, shaft power, and fuel consumption prediction.

Vessel	Train Set Instances	Test Set Instances	Train Dates (Start → End)	Test Dates (Start → End)	Dry Docking Date
A-6	12,758	21,930	2023-01-09 → 2024-01-15	2024-01-15 → 2025-09-01	14/12/2022
A-8	18,145	8,663	2020-04-18 → 2020-10-13	2020-10-18 → 2021-09-14	14/04/2022
A-10	20,608	91,664	2020-05-11 → 2022-03-27	2022-04-22 → 2025-03-29	23/05/2023
A-12	23,186	96,258	2020-09-21 → 2023-02-19	2023-02-22 → 2025-05-07	09/11/2023
A-13	22,970	70,077	2020-04-20 → 2021-11-20	2021-11-22 → 2025-05-02	14/08/2023

TABLE II : Number of instances and data periods for each vessel in training and test sets, including dry docking dates.

Panel of Figure 4 — Fig. 3 : Comparison of actual vs predicted performance metrics for A-6 Vessel.

Vessel	Method	shaft_rpm			shaft_power			fuel_consumed
		MAE	RMSE	R 2	MAE	RMSE	R 2	MAE	RMSE	R 2
A-6	Polynomial	2.34	2.97	88.37	12.39	15.80	60.34	17.52	22.98	2.01
A-6	MLP	6.74	8.19	11.91	13.55	17.93	48.93	25.18	31.24	-81.04
	KAN	10.42	11.61	77	33.52	40.47	61.51	26.01	25.25	71.42
	PI-KAN	1.88	3.33	90.79	8.89	10.96	77.27	11.24	13.98	72.47
A-8	Polynomial	2.34	3.02	85.85	13.98	18.18	34.08	16.42	21.05	-7.89

TABLE III : Benchmark performance on the test set (relative errors, overall values) across different targets. MAE: Mean Absolute Error, RMSE: Root Mean Squared Error, R 2 : Coefficient of Determination.

Vessel	Departure (UTC)	Duration	Samples	PI-KAN		Traditional
				ME% (signed)	MAPE%	ME% (signed)	MAPE%
A-1	2020-07-28 18:30:00	10 days 02:45	944	22.41	22.63	12.00	–
A-11	2020-11-29 03:15:00	13 days 05:30	179	5.55	10.17	-8.00	–

TABLE IV : Performance analysis of adversarial weather conditions (fuel_consumption). Test voyage window = departure for a duration of travel in that period. (All other voyages (different voyage_ids) go into training). Traditional model results are added for comparison.

Fig. 4 : Adversarial weather response (A–1 vs. A–11): vessel speed as a function of wave height. A–11 reduces speed at higher waves, while A–1 holds a near-constant setpoint. This difference aligns with Table IV : lower error on A–11 (ME=+5.55%, MAPE=10.17%) versus higher error on A–1 (ME=+22.41%, MAPE=22.63%).

Fig. 5 : Learned univariate feature transformation functions of the KAN model for shaft power prediction (epoch 76). Each subplot shows how an input feature is nonlinearly mapped before combination in the network. The transformations reflect physically meaningful patterns, including cubic-like dependence on ship speed through water ( stw_ms ), cosine-like wind and wave directional effects, and localized nonlinear responses to swell and sea depth. The top-left subplot displays the training loss curve converging after $\sim$ 60 epochs.

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

Orfeas Bourchas, George Papalambrou

Laboratory of Marine Engineering N.T.U.A.

Physics-Informed ML Residual Learning

Open paper AskPDF arXiv

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.

Abstract

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

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