A physics–machine learning hybrid model for ignition-delay estimation considering negative temperature coefficient behavior

Accurate ignition-delay estimation is essential for combustion control in diesel engines because ignition delay strongly influences engine efficiency and emissions. However, ignition delay exhibits highly nonlinear temperature dependence due to the presence of the negative temperature coefficient (NTC) region, where ignition delay increases with increasing temperature. Conventional ignition-delay models based on a single Arrhenius-type temperature dependence cannot adequately represent this non-monotonic behavior. In this study, a physics–machine learning hybrid model is proposed to estimate ignition delay while accounting for NTC-like behavior. The proposed approach extends a previously proposed Model-Weighting Neural Network (MWNN) framework by introducing ignition-delay basis models with different apparent temperature dependences. In this framework, multiple physics-based ignition-delay basis models are prepared, and their contributions are determined through state-dependent weighting by a neural network according to the in-cylinder gas state and engine operating conditions. The proposed method was evaluated using actual engine test data obtained from a production four-cylinder common-rail diesel engine operated on an engine test bench. Compared with a conventional single Arrhenius-type model, the proposed MWNN model reduced the RMSE from 0.751° to 0.464° and improved the coefficient of determination R 2 from 0.660 to 0.870 over the entire test dataset. Furthermore, the introduction of NTC-related basis models improved estimation accuracy particularly in the low-temperature region where NTC-like behavior was observed. Analysis of model contributions indicates that multiple ignition-delay basis models cooperatively represent ignition-delay behavior across different temperature regimes. These results indicate that the proposed method can more appropriately represent NTC-like non-monotonic temperature dependence over the examined operating range while retaining a control-oriented algebraic model structure and high estimation accuracy.