  Joint probability density function (PDF)-based models in turbulent combustion
provide direct closure for turbulence-chemistry interactions. The joint PDFs
capture the turbulent flame dynamics at different spatial locations and hence
it is crucial to represent them accurately. The jointPDFs are parameterized on
the unconditional means of thermo-chemical state variables, which can be high
dimensional. Thus, accurate construction of joint PDFs at various spatial
locations may require an exorbitant amount of data. In a previous work, we
introduced a framework that alleviated data requirements by constructing joint
PDFs in a lower dimensional space using principal component analysis (PCA) in
conjunction with Kernel Density Estimation (KDE). However, constructing the
principal component (PC) joint PDFs is still computationally expensive as they
are required to be calculated at each spatial location in the turbulent flame.
In this work, we propose the concept of a generalized joint PDF model using the
Deep Operator Network (DeepONet). The DeepONet is a machine learning model that
is parameterized on the unconditional means of PCs at a given spatial location
and discrete PC coordinates and predicts the joint probability density value
for the corresponding PC coordinate. We demonstrate the accuracy and
generalizability of the DeepONet on the Sandia flames, D, E and F. The DeepONet
is trained based on the PC joint PDFs observed inflame E and yields excellent
predictions of joint PDFs shapes at different spatial locations of flamesD and
F, which are not seen during training
