# A Causal Graph-Enhanced Gaussian Process Regression for Modeling Engine-out NOx

arXiv: https://arxiv.org/abs/2410.18424 · HTML: https://mlsanity.com/paper/2410.18424/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 35 references that are in the map.
- First posted on arXiv 24 Oct 2024.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [Gated Graph Sequence Neural Networks](https://mlsanity.com/paper/1511.05493/) (2015)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Manifold Gaussian Processes for Regression](https://mlsanity.com/paper/1402.5876/) (2014)
- [Directed Acyclic Graph Neural Networks](https://mlsanity.com/paper/2101.07965/) (2021)
- [Relating Graph Neural Networks to Structural Causal Models](https://mlsanity.com/paper/2109.04173/) (2021)
- [DKL-KAN: Scalable Deep Kernel Learning using Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2407.21176/) (2024)
- [Leveraging Gated Recurrent Units for Iterative Online Precise Attitude Control for Geodetic Missions](https://mlsanity.com/paper/2405.15159/) (2024)

## Built on by (most cited first, 1 of 1)

- [Bayesian Calibration of Engine-out NOx Models for Engine-to-Engine Transferability](https://mlsanity.com/paper/2511.18178/) (2025)

## Abstract (arXiv)

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.