Hybrid symbolic regression framework for digital twin modeling of gas internal combustion engines

This study introduces a real-time, control-oriented digital twin framework for hydrogen internal combustion engines (H2ICEs) identified from production actuator command signals and a measured torque target. The objective is to reproduce the torque-path input–output behavior seen by the engine control unit under transient operation, enabling controller prototyping (including predictive strategies that rely on torque prediction), hardware-in-the-loop evaluation, and fault detection under a fixed calibration. A recursive symbolic regression framework is developed by integrating the Robust Iterated Local Search with Ordinary Least Squares (RILS-ROLS) and Operon algorithms within a Nonlinear Finite Impulse Response (NFIR) configuration. The method iteratively refines model accuracy while maintaining low computational cost and interpretability. Using experimental data from a heavy-duty 13-liter spark-ignited H2ICE operated under the World Harmonized Transient Cycle (WHTC), the recursive NFIR model improved its root mean square error ( RMSE ) by 47.2% over 27 iterations, achieving a coefficient of determination ( R 2 ) of 0.9892 and an RMSE of 52.33 [N .m] (newton-meter) with a final model size of 632 [B] (bytes). To capture residual dynamics, a hybrid modeling strategy was employed in which the NFIR model served as a baseline predictor and black-box regressors were trained on its residuals. The best configuration, HNFIR TabPFN , reached an R 2 of 0.9981 and an RMSE of 21.90 [N .m], improving accuracy at the expense of model size of 101 [MB] (megabytes). This trade-off highlights that while the standalone NFIR model is ideal for embedded real-time deployment, the hybrid models are better suited for high-fidelity offline simulation within the studied operating domain when maximum predictive precision is required.