# Deep learning for presumed probability density function models

## TL;DR

This paper explores machine learning techniques, including deep neural networks and variational autoencoders, to develop more accurate presumed PDF models for large eddy simulations of reacting flows, outperforming traditional analytical models.

## Contribution

It introduces ML-based presumed PDF models that are more accurate, faster, and require fewer parameters than existing analytical models for reacting flow simulations.

## Key findings

- Deep learning models are three times more accurate than analytical beta-beta PDFs.
- ML models are as accurate as random forests but with fewer parameters.
- Models generalize well to different flow regions and are evaluated using Jensen-Shannon divergence.

## Abstract

In this work, we use ML techniques to develop presumed PDF models for large eddy simulations of reacting flows. The joint sub-filter PDF of mixture fraction and progress variable is modeled using various ML algorithms and commonly used analytical models. The ML algorithms evaluated in the work are representative of three major classes of ML techniques: traditional ensemble methods (random forests), deep learning (deep neural networks), and generative learning (variational autoencoders). The first two algorithms are supervised learning algorithms, and the third is an unsupervised learning algorithm. Data from direct numerical simulation of the low-swirl burner (Day et al. 2012) are used to develop training data for sub-filter PDF models. Models are evaluated on predictions of the sub-filter PDFs as well as predictions of the filtered reaction rate of the progress variable, computed through a convolution of the sub-filter PDF and the conditional means of the reaction rate. This a-priori modeling study demonstrates that deep learning models for presumed PDF modeling are three times more accurate than analytical beta-beta PDF models. These models are as accurate as random forest models while using five times fewer trainable parameters and being 25 times faster for inference. We illustrate how models generalize to other regions of the flow and develop criteria based on the Jensen-Shannon divergence to quantify the performance of a model on new data.