  We develop a data-driven deep neural operator framework to approximate
multiple output states for a diesel engine and generate real-time predictions
with reasonable accuracy. As emission norms become more stringent, the need for
fast and accurate models that enable analysis of system behavior have become an
essential requirement for system development. The fast transient processes
involved in the operation of a combustion engine make it difficult to develop
accurate physics-based models for such systems. As an alternative to physics
based models, we develop an operator-based regression model (DeepONet) to learn
the relevant output states for a mean-value gas flow engine model using the
engine operating conditions as input variables. We have adopted a mean-value
model as a benchmark for comparison, simulated using Simulink. The developed
approach necessitates using the initial conditions of the output states to
predict the accurate sequence over the temporal domain. To this end, a
sequence-to-sequence approach is embedded into the proposed framework. The
accuracy of the model is evaluated by comparing the prediction output to ground
truth generated from Simulink model. The maximum $\mathcal L_2$ relative error
observed was approximately $6.5\%$. The sensitivity of the DeepONet model is
evaluated under simulated noise conditions and the model shows relatively low
sensitivity to noise. The uncertainty in model prediction is further assessed
by using a mean ensemble approach. The worst-case error at the $(\mu +
2\sigma)$ boundary was found to be $12\%$. The proposed framework provides the
ability to predict output states in real-time and enables data-driven learning
of complex input-output operator mapping. As a result, this model can be
applied during initial development stages, where accurate models may not be
available.
