Progress Variable Variance and Filtered Rate Modelling Using Convolutional Neural Networks and Flamelet Methods

Abstract

A purely data-driven modelling approach using deep convolutional neural networks is discussed in the context of Large Eddy Simulation (LES) of turbulent premixed flames. The assessment of the method is conducted a priori using direct numerical simulation data. The network has been trained to perform deconvolution on the filtered density and the filtered density-progress variable product, and by doing so obtain estimates of the un-filtered progress variable field. A filtered function of the progress variable can then be approximated on the LES mesh using the deconvoluted field. This new strategy for tackling turbulent combustion modelling is demonstrated with success for both the sub-grid scale progress variable variance and the filtered reaction rate, using flamelet methods, two fundamental ingredients of premixed turbulent combustion modelling.

Appendix: Network structure

A CNN usually consists of convolutional and sub-sampling layers accompanied by fully connected layers. Each of the convolutional layers of the CNN can have K number of filters (kernels). An essential aspect of the CNN is the size of the filters, which can identify the locally connected structure and in-turn convoluted with the input to create K feature maps. Each of the generated K feature maps can then be sub-sampled using min or max pooling for a defined region, typically between 2-5 points. Furthermore, another important part of the CNN is the addition of a bias parameter and the application of a linear or non-linear activation function for each feature map. The use of bias and the activation function can be applied either before or after the sampling of the feature maps. A mean squared error was used as an error measure between the predicted and target variables, and the training conducted in the standard approach using the back-propagation algorithm [57].

Figure 8 shows the structure of the network used for the deconvolution. The network consists of a series of convolution, normalisation and application of an activation function layers, in that order. The input layer consists of the set of 11³ points holding the filtered values in the halo cube for a given point on the LES mesh. The output from the first convolution layer is an 8³ set of features for each of the 256 kernels used. The output from a convolution layer may have values which differ by a large amount, and are batch-normalised first before applying the activation function. In this process, the data are normalised by subtracting the batch mean, and dividing it by the standard deviation of the set. Following this, a leaky Rectified Linear Unit activation function (RELU) [35] is applied to the extracted features (thresholding)-in our case we have used f(x) = x for x > 0, and f(x) = 0.3x otherwise. A big advantage of this type of activation function is that is computationally faster to evaluate when compared for example to a sigmoid function which involves the evaluation of an exponential term. This speeds up the training process but also helps convergence [36]. In the second convolution layer, the output from the RELU is convoluted using 128 kernels, resulting in an output set of 5³ features for each of the 128 kernels. The process is repeated with convolutional, RELU and normalisation layers. During the training phase, the weights of all kernels in each convolutional layer are adjusted so as to minimise the mean-squared error between the deconvoluted and actual field. In the end, a total of 32 features are extracted which are connected to a single node having a linear activation function and resulting in a single output namely the deconvoluted field. The total size of the training data for each case depends on the size of the LES mesh. In particular, for an LES mesh having N_x,N_y,N_z points in space and N_t datasets in time, the total size of the training data is $N_{x} \cdot N_{y} \cdot N_{z} \cdot N_{t} \cdot {N_{h}^{3}}$ where N_h is the size of the halo cube around each point on the LES mesh. Therefore depending on the size of the DNS database the training data size can become significant.

Fig. 8

The structure of the convolutional network