Bayesian Calibration of Engine-out NOx Models for Engine-to-Engine Transferability
Shrenik Zinage, Peter Meckl, Ilias Bilionis
TL;DR
The paper tackles the challenge of transferring engine-out NOx predictive models across engines by introducing a Bayesian calibration framework that augments a pre-trained Gaussian Process with engine-specific sensor biases inferred via Approximate Bayesian Computation. Using a KS-based distance and adaptive thresholding, the method yields posterior predictive distributions for unseen engines without retraining the GP. The approach demonstrates improved predictive accuracy and robust uncertainty quantification across multiple Cummins diesel engines and operating cycles, highlighting enhanced transferability and reduced calibration costs. This work offers a scalable pathway for fleet-wide, data-driven NOx prediction with principled handling of engine-to-engine variability and sensor discrepancies.
Abstract
Accurate prediction of engine-out NOx is essential for meeting stringent emissions regulations and optimizing engine performance. Traditional approaches rely on models trained on data from a small number of engines, which can be insufficient in generalizing across an entire population of engines due to sensor biases and variations in input conditions. In real world applications, these models require tuning or calibration to maintain acceptable error tolerance when applied to other engines. This highlights the need for models that can adapt with minimal adjustments to accommodate engine-to-engine variability and sensor discrepancies. While previous studies have explored machine learning methods for predicting engine-out NOx, these approaches often fail to generalize reliably across different engines and operating environments. To address these issues, we propose a Bayesian calibration framework that combines Gaussian processes with approximate Bayesian computation to infer and correct sensor biases. Starting with a pre-trained model developed using nominal engine data, our method identifies engine specific sensor biases and recalibrates predictions accordingly. By incorporating these inferred biases, our approach generates posterior predictive distributions for engine-out NOx on unseen test data, achieving high accuracy without retraining the model. Our results demonstrate that this transferable modeling approach significantly improves the accuracy of predictions compared to conventional non-adaptive GP models, effectively addressing engine-to-engine variability and improving model generalizability.