Physics-constrained and uncertainty-aware Bayesian optimisation of hydrogen–diesel dual-fuel compression ignition engines for performance–emission trade-off analysis

Hydrogen–diesel dual-fuel compression-ignition (CI) engines can improve thermal efficiency while reducing reliance on conventional fossil fuels. However, the strong trade-off between efficiency and nitrogen oxide (NO _x ) emissions makes the identification of optimal operating conditions challenging, particularly when experimental data are limited. This study investigates the performance–emission characteristics of a variable-compression-ratio hydrogen–diesel dual-fuel engine using a physics-constrained Gaussian Process Regression (PC-GPR) model integrated with uncertainty-aware multi-objective Bayesian optimisation. Experiments were conducted over compression ratios of 12–18, torque levels of 5–20 N·m and hydrogen energy fractions of 0.27–0.40, generating a dataset comprising both dual-fuel and diesel-baseline operation. Peak in-cylinder temperature and cumulative heat release obtained from combustion analysis were incorporated to improve physical consistency, and 11 combustion-based constraints were used to verify surrogate behaviour. Model performance was assessed using leave-one-out cross-validation. The PC-GPR model achieved cross-validated R ² values of 0.97 for brake thermal efficiency (BTE) and 0.91 for NO _x emissions on the combined dataset, outperforming eight alternative machine-learning models. SHAP analysis identified torque as the dominant driver of brake thermal efficiency and peak in-cylinder temperature as the leading thermal driver of NO _x emissions. Multi-objective Bayesian optimisation using qLogNEHVI enabled continuous Pareto-front improvement and concentrated the search where the surrogate is well determined: the mean predictive standard deviation in brake thermal efficiency at the returned solutions is 0.26%, against 0.96% for a deterministic NSGA-II search that is insensitive to prediction confidence. A reliability-aware Pareto analysis identified an operating region at a compression ratio of 13.5, torque of 15.6 N·m and hydrogen energy fraction of 0.30, corresponding to a predicted BTE of 26.3% and NO _x emissions of 385 ppm. At the nearest measured condition, all five predicted responses fell within their 95% prediction intervals, with a mean deviation of 8.2%. The proposed approach offers a reliable and data-efficient framework for hydrogen–diesel engine calibration and optimisation.