[Paper Review] Modeling diesel output particulate matter as the Ornstein-Uhlenbeck process

Maxwell Bolt, Alex Alberts+3 more|arXiv (Cornell University)|Mar 12, 2026

Advanced Combustion Engine Technologies0 citations

TL;DR

The paper proposes a stochastic Ornstein-Uhlenbeck (OU) process model with time-varying drift and volatility, parameterized by engine states, to predict engine-out diesel PM with calibrated uncertainty. It trains via maximum likelihood and validates on EPA drive cycles, outperforming some baselines in cumulative PM prediction.

ABSTRACT

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.

Motivation & Objective

• Motivate the need for a stochastic, computationally light PM model in transient diesel engine operation.
• Introduce an OU-process-based framework with time-varying drift and diffusion tuned to engine states.
• Develop a tractable training approach using likelihood-based optimization and Fokker-Planck transition densities.
• Demonstrate model verification with synthetic data and validate on EPA-driven test cycles.
• Assess predictive performance and uncertainty quantification against benchmarks.

Proposed method

• Model PM dynamics as a modified Ornstein-Uhlenbeck SDE: dX_t = λ(μ_t − X_t) dt + σ_t dW_t with time-varying μ_t and σ_t.
• Parameterize μ_t and σ_t as linear functions of 16 engine state/control variables, with λ as a constant mean-reversion rate.
• Train parameters θ via maximum likelihood using the transition Gaussian density derived from the Fokker-Planck formulation.
• Compute transition mean m_θ(t) and variance V_θ(t) for piecewise-constant μ_t and σ_t and use them to form the likelihood.
• Transform data with a piecewise log-linear normalization to stabilize training.
• Deploy by simulating sample paths from the learned OU process and evaluating uncertainty with PIT and KS tests.

Experimental results

Research questions

• RQ1 Can an OU-driven stochastic model with time-varying drift and volatility capture transient diesel PM generation?
• RQ2 How well does the linear-in-engine-states parameterization of μ_t and σ_t perform compared to alternative baselines?
• RQ3 What is the predictive quality and uncertainty calibration of the OU model across EPA drive cycles and test datasets?
• RQ4 Does augmenting training data with specific cycles improve out-of-distribution performance such as RMCSET?
• RQ5 How does the OU model compare to Gaussian processes and neural networks in predicting PM dynamics under transient conditions?

Key findings

• The OU-based model with time-varying drift and volatility can predict cumulative PM across time on most test cycles.
• KS statistics vary by dataset, e.g., FTP Low NOx (0.085), FTP Nominal NOx (0.290), FTP High NOx (0.246), STC Nominal NOx (0.134), STC High NOx (0.057), with poorer performance on RMCSET (0.482).
• The model tends to overpredict PM in the right tail for some cycles, as shown by Q–Q plots of PIT values.
• Appending part of the RMCSET data to training improved performance on RMCSET and several other datasets (reduced KS in several cases).
• Compared against two Gaussian processes (standard and deep-kernel) and several neural-network baselines, the CNN deep-kernel GP underperformed the standard GP, suggesting PM dynamics are more Markovian than NOx in this context.
• The framework enables sampling 10,000 paths to characterize the predictive distribution and uncertainty of PM over time.