A Causal Graph-Enhanced Gaussian Process Regression for Modeling Engine-out NOx
This work addresses the need for robust diagnostic models for engine emissions monitoring, but it is incremental as it builds on existing Gaussian process methods with domain-specific modifications.
The paper tackled the problem of accurately predicting engine-out NOx emissions for diesel engines by developing a probabilistic model using Gaussian process regression, resulting in improved predictive performance with enhancements from a deep kernel and causal graph incorporation.
The stringent regulatory requirements on nitrogen oxides (NOx) emissions from diesel compression ignition engines require accurate and reliable models for real-time monitoring and diagnostics. Although traditional methods such as physical sensors and virtual engine control module (ECM) sensors provide essential data, they are only used for estimation. Ubiquitous literature primarily focuses on deterministic models with little emphasis on capturing the various uncertainties. The lack of probabilistic frameworks restricts the applicability of these models for robust diagnostics. The objective of this paper is to develop and validate a probabilistic model to predict engine-out NOx emissions using Gaussian process regression. Our approach is as follows. We employ three variants of Gaussian process models: the first with a standard radial basis function kernel with input window, the second incorporating a deep kernel using convolutional neural networks to capture temporal dependencies, and the third enriching the deep kernel with a causal graph derived via graph convolutional networks. The causal graph embeds physics knowledge into the learning process. All models are compared against a virtual ECM sensor using both quantitative and qualitative metrics. We conclude that our model provides an improvement in predictive performance when using an input window and a deep kernel structure. Even more compelling is the further enhancement achieved by the incorporation of a causal graph into the deep kernel. These findings are corroborated across different verification and validation datasets.