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Joint probability density function (PDF)-based models in turbulent combustionprovide direct closure for turbulence-chemistry interactions. The joint PDFscapture the turbulent flame dynamics at different spatial locations and henceit is crucial to represent them accurately. The jointPDFs are parameterized onthe unconditional means of thermo-chemical state variables, which can be highdimensional. Thus, accurate construction of joint PDFs at various spatiallocations may require an exorbitant amount of data. In a previous work, weintroduced a framework that alleviated data requirements by constructing jointPDFs in a lower dimensional space using principal component analysis (PCA) inconjunction with Kernel Density Estimation (KDE). However, constructing theprincipal component (PC) joint PDFs is still computationally expensive as theyare 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 theDeep Operator Network (DeepONet). The DeepONet is a machine learning model thatis parameterized on the unconditional means of PCs at a given spatial locationand discrete PC coordinates and predicts the joint probability density valuefor the corresponding PC coordinate. We demonstrate the accuracy andgeneralizability of the DeepONet on the Sandia flames, D, E and F. The DeepONetis trained based on the PC joint PDFs observed inflame E and yields excellentpredictions of joint PDFs shapes at different spatial locations of flamesD andF, which are not seen during training

Generalized Joint Probability Density Function Formulation inTurbulent Combustion using DeepONet