Deep Learning for Presumed Probability Density Function Models

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49 Scopus Citations


In this work, we use machine learning (ML) techniques to develop presumed probability density function (PDF) models for large eddy simulations (LES) 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 network (DNN)s), and generative learning (conditional variational autoencoder (CVAE)). The first two algorithms are supervised learning algorithms, and the third is an unsupervised learning algorithm. Data from direct numerical simulation (DNS) of the low-swirl burner[1] are used to develop training data for sub-filter PDF models. Models are evaluated on predictions of the sub-filter PDF as well as predictions of the filtered reaction rate of the progress variable, computed through an integral of the product 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 β–β PDF models and linear regression models. These models are as accurate as random forest models while using five times fewer trainable parameters and being 25 times faster for inference. In this work, conditional unsupervised learning did not present additional advantages beyond supervised learning with a feed-forward neural network. 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.

Original languageAmerican English
Pages (from-to)436-450
Number of pages15
JournalCombustion and Flame
StatePublished - Oct 2019

Bibliographical note

Publisher Copyright:
© 2019 The Combustion Institute

NREL Publication Number

  • NREL/JA-2C00-73038


  • Large eddy simulation
  • Low-swirl burner
  • Machine learning
  • Presumed probability density function
  • β–β PDF


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