Gearbox Bearing Crack Growth Prognostics with Physics-Informed Machine Learning and Uncertainty Quantification

Abstract. This paper introduces the eXtreme Theory of Functional Connections (X-TFC), a physics-informed machine learning algorithm, and tailors it to estimate the remaining useful life (RUL) of wind turbine gearbox bearings experiencing fatigue crack growth. Unlike purely data-driven methods, X-TFC embeds a physics model, based on the Head’s theory in this work, into its training objective. The core of X-TFC is a random-projection single-layer neural network trained via Extreme Learning Machine, which requires only limited damage progression data and solves for output weights with a least-squares optimization algorithm. A composite loss function balances the network’s fit to observed degradation data against the residuals of the governing crack-growth differential equation, ensuring the learned damage trajectory remains physically plausible. When applied to a vibration-based health-index (HI) dataset measured during the growth of a crack on the inner ring of a high-speed bearing in a wind turbine gearbox (Bechhoefer and Dubé, 2020), X-TFC achieves near-zero prediction bias. Even when trained on only the first 10–20 % of the damage progression data, its predictions remain monotonic and smooth, delivering high prognosability and trendability. To quantify the epistemic uncertainty, we employ a Monte Carlo ensemble of independently initialized X-TFC models trained on noise-perturbed data, which yields confidence intervals around each RUL estimate. This approach provides confidence intervals around each RUL estimate, capturing both model-parameter and epistemic uncertainty. In ad- dition to a vibration-based HI, we demonstrate that the proposed framework can be directly applied to a SCADA data-based HI (Eftekhari Milani et al., 2025) measured during similar wind turbine gearbox bearing crack faults, preserving its accuracy and interpretability. This extension shows the versatility of our approach, which is applicable to bearings of multiple gearbox manufacturers, models and ratings using only SCADA data. By integrating domain knowledge with machine learning, X-TFC offers a rapid, reliable tool for crack prognostics. Its adaptability to other bearing failure modes, such as pitch-bearing ring cracks, positions X-TFC as a powerful enabler of data-driven, physics-informed asset management in the wind energy sector and beyond.