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. Production engines typically do not have calibrated, time-resolved sensors for continuous engine-out PM mass estimation, so manufacturers rely on predictive models using available engine state measurements. 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. Using high-frequency experimental PM measurements from transient engine tests, 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 and is defined as the solution to 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, parameterized 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 PM mass output across time.