Physics-Informed Deep Learning for Bearing Remaining Useful Life: Adaptive Paris-Law Regularization with Gaussian Processes

Abstract Accurate estimation of remaining useful life is essential for predictive maintenance and for preventing costly unplanned downtime in industrial assets. However, state-of-the-art deep learning models often behave as black boxes, offer limited interpretability, and can degrade under noise, data scarcity, or distributional shifts. Many physics-informed neural networks address this by embedding governing equations, but they typically fix physical coefficients a priori, which can bias predictions when units, loads, or operating conditions change. This work presents an adaptive physics-informed deep learning framework that couples sequence learning with a fracture mechanics-based structure. For each degradation sequence, Paris-law parameters are estimated by combining Gaussian process regression on finite-difference rates with constrained nonlinear least squares. This procedure yields sequence-specific and physically meaningful regularization. The Paris-induced decay is implemented as a soft residual using a Huber penalty and an annealed curriculum. The strength of this residual term, along with the model width and learning rate, is selected by Bayesian optimization to balance data fit and physical plausibility. Predictive uncertainty is quantified using deep ensembles with optional split-conformal calibration, and a physics consistency score is defined to measure adherence to the underlying deterioration law. Experiments on the FEMTO PRONOSTIA bearing dataset show that treating physics as a learnable, data-adaptive prior, rather than a static constraint, produces accurate, interpretable, and robust remaining useful life predictions. This design makes the proposed approach suitable for deployment in diverse industrial environments.