Modeling diesel output particulate matter as the Ornstein-Uhlenbeck process

Diesel engine particulate matter (PM) is one of the most challenging emission constituents to predict. As engines become cleaner and emissions levels drop, manufacturers need reliable methods to quantify the PM generated by production engines. Due to the inaccuracy of commercial-grade sensors, they turn to predictive models to accurately estimate PM. In practice, this requires a computationally inexpensive model that provides PM estimates with calibrated uncertainty. Complex, multiscale physics make mechanistic models intractable and traditional data-driven methods struggle in transient drive cycles due to the stochastic nature of PM generation. Leveraging recent innovations in PM measurement technology, we introduce a novel PM model based on the Ornstein-Uhlenbeck (OU) process. The OU process is a mean-reverting stochastic process commonly used in financial modeling, now being explored for engineering applications, and can be described as a stochastic differential equation (SDE). We modify the OU process by parameterizing the terms of the SDE as functions of the engine state, which are then fit with a maximum likelihood estimate. In a synthetic example, we verify the ability of our model to learn a time-varying, parametrized OU process. We then train the model using real experimental data designed to dynamically cover the engine operating space and test the trained model on EPA-regulated drive cycles. For most drive cycles, we find the method accurately predicts cumulative output of PM across time.