Hybrid physics–machine learning approach combining Gaussian process regression and Sines-style invariants for multiaxial fatigue life prediction of S355 steel

• A baseline-anchored hybrid physics–machine learning framework is developed for multiaxial fatigue-life prediction of S355 steel. • Sines–Basquin and Fatemi–Socie baselines are compared with standalone Gaussian process regression and three hybrid strategies. • Residual hybridization (physics + GPR correction) yields the best accuracy: R² = 0.955 and RMSE = 0.126 in log 10 ( N ). • Hybridization cuts RMSE by about 47% versus Sines–Basquin and reduces nominal 95% interval width by about 40%. • The proposed workflow supports interpretable and reliability-oriented prediction under proportional loading. • The workflow provides uncertainty-aware predictions suitable for reliability-oriented fatigue assessment under proportional loading. This study develops a hybrid physics–machine learning framework for multiaxial fatigue-life prediction of S355 structural steel under proportional (in-phase) loading. Two classical criteria are first established as physics baselines: an invariant-based Sines Basquin formulation and a critical-plane Fatemi Socie (FS) formulation, each calibrated in log 10 ( N ) space using Basquin type regression. Their predictive capability is benchmarked against a standalone Gaussian process regression (GPR) model trained on stress-state descriptors. The best-performing physics baseline is then integrated with GPR to form three hybrid variants, improving accuracy while preserving an interpretable mechanics-driven reference trend. All models are trained and evaluated on a curated dataset of 133 multiaxial fatigue tests, with performance assessed using R², MAE, and RMSE in log 10 ( N ). Uncertainty quality is evaluated through 95% prediction intervals to support reliability critical decision making. Recent studies have applied GPR based approaches to multiaxial fatigue prediction with uncertainty quantification, often in fully data driven settings (e.g., neural predictors coupled with GPR; Gao et al., 2026) or through empirical corrections that do not explicitly embed classical multiaxial fatigue criteria. In contrast, the present framework retains the physics criterion explicitly and uses GPR to learn the remaining discrepancy, yielding uncertainty-aware predictions anchored to established fatigue descriptors. Across the dataset, hybridization consistently outperforms both physics-only and purely data-driven baselines. The top-performing hybrid configuration reaches R² = 0.955, MAE = 0.098, and RMSE = 0.126 in log 10 ( N ), while producing comparatively tight and well-calibrated 95% prediction intervals.