Solve 2D Navier stokes Equation Deep Learning

Hi I want solve 2D NS with the help of Machine learning. I want to modify the use case of the Burges Equation(BE) to 2d NS. Here is the link of the UC of how to solve the BE with deep learning.

https://ch.mathworks.com/help/deeplearning/ug/solve-partial-differential-equations-using-deep-learning.html

How ever I need to extract U and V (x and y component of the Velocity) and Presure P from the Model to solve the 2D-NS.

function dlU = model(parameters,dlX,dlT)

dlXT = [dlX;dlT];
numLayers = numel(fieldnames(parameters));

% First fully connect operation.
weights = parameters.fc1.Weights;
bias = parameters.fc1.Bias;
dlU = fullyconnect(dlXT,weights,bias);

% tanh and fully connect operations for remaining layers.
for i=2:numLayers
    name = "fc" + i;

    dlU = tanh(dlU);

    weights = parameters.(name).Weights;
    bias = parameters.(name).Bias;
    dlU = fullyconnect(dlU, weights, bias);
end

end

First, I have to add y to the Input data, to have (dlX,dlY,dlT) as Inputs. But I'am not quite sure how to adopt the Model function to get V and P?

Any Idea is welcome.

BR,

Chris

[CSCh]

If I try to put more Output Parameters in the Model function I got the error "too many outputs, referring to the fullyconnect command. Any Idea?

function [dlU,dlV, dlP] = model(parameters,dlX,dlY,dlT)

dlXT = [dlX;dlY;dlT];
numLayers = numel(fieldnames(parameters));

% First fully connect operation.
weights = parameters.fc1.Weights;
bias = parameters.fc1.Bias;
[dlU,dlV, dlP] = fullyconnect(dlXT,weights,bias);

% tanh and fully connect operations for remaining layers.
for i=2:numLayers
    name = "fc" + i;

    dlU = tanh(dlU);
     dlV = tanh(dlV);
     dlP = tanh(dlP);
    weights = parameters.(name).Weights;
    bias = parameters.(name).Bias;
    [dlU,dlV, dlP] = fullyconnect(dlU,dlV,dlP, weights, bias);
    %dlP = fullyconnect(dlP, weights, bias);
end

end

Answer 1

[Andreas Apostolatos]

Dear Chris,

Function 'fullyconnect' returns only a single output argument, see the following documentation page for more information,

https://www.mathworks.com/help/deeplearning/ref/dlarray.fullyconnect.html

and that is the cause of the error you are receiving when trying to include additional output arguments. Function 'fullyconnect' is used to sum all the weighted input data and apply a bias, as mentioned in the latter documentation page.

In that sense, the output argument 'dlU' should contain all the necessary components for the problem at hand and there should not be any additional output argument.

What needs to be modified in this case is the layer architecture so that it allows for three input channels (X-, Y-components and time) while it subsequently it also allows for three output channels (U-, V-components of the velocity and pressure). To account for these three input channels, the following lines of code for the shipping 1D Burger's example,

% Initialize the parameters for the first fully connect operation. The first fully connect operation has two input channels.
% parameters = struct;

sz = [numNeurons 2];
parameters.fc1.Weights = initializeHe(sz,2);
parameters.fc1.Bias = initializeZeros([numNeurons 1]);

need to be changed to,

% Initialize the parameters for the first fully connect operation. The first fully connect operation has two input channels.
% parameters = struct;

sz = [numNeurons 3];
parameters.fc1.Weights = initializeHe(sz,2);
parameters.fc1.Bias = initializeZeros([numNeurons 1]);

In this way, the 'dlarray' containing the spatial coordinates of the collocation points that are used for the training should no longer be defined as in the shipping 1D Burger's example, namely,

% Convert the initial and boundary conditions to dlarray. For the input data points, specify format with dimensions 'CB' (channel, batch).
dlX0 = dlarray(X0,'CB');

but as follows,

% Convert the initial and boundary conditions to dlarray. For the input data points, specify format with dimensions 'CB' (channel, batch).
dlX0 = dlarray([X0; Y0],'CB');

where 'Y0' is a vector containing all the Y-components of your collocation points. This vector should be created in the same manner as vector 'X0' and all other 'dlarray' objects containing such spatial information should be extended accordingly. This is herein necessary since the input layer of your deep neural network expects three input channels.

Next, the output layer of your deep neural network should be changed in order to account for three output channels (U-, V-components of the velocity and pressure) instead of just one channels as for the shipping 1D Burger's example. In this way, the following lines of code of the shipping example,

% Initialize the parameters for the final fully connect operation. The final fully connect operation has one output channel.
sz = [1 numNeurons];
numIn = numNeurons;
parameters.("fc" + numLayers).Weights = initializeHe(sz,numIn);
parameters.("fc" + numLayers).Bias = initializeZeros([1 1]);

needs to be changed to the following,

% Initialize the parameters for the final fully connect operation. The final fully connect operation has one output channel.
sz = [3 numNeurons];
numIn = numNeurons;
parameters.("fc" + numLayers).Weights = initializeHe(sz,numIn);
parameters.("fc" + numLayers).Bias = initializeZeros([3 1]);

where both the number of channels for the input to the final fully connected layer and the number of biases needs to be modified accordingly.

Having made these changes, there should be applied no actual change in the function 'model', as the fully connected operations should already result into three outputs. The output of the last 'fullyconnect' call in the following loop (i.e. 'i = numLayers') within the 'model' function,

% tanh and fully connect operations for remaining layers.
for i=2:numLayers
     name = "fc" + i;

    dlU = tanh(dlU);

    weights = parameters.(name).Weights;
    bias = parameters.(name).Bias;
    dlU = fullyconnect(dlU, weights, bias);
end

should herein result into a 3(C) × 1000(B) (instead of a 1(C) x 1000(B)) 'dlarray' object, where the first channel corresponds to the U-components of the velocity field, the second channel corresponds to the V-components of the velocity field and finally the third channel corresponds to the pressure field.

These are in principle the structural changes that need to be applied in order to have the appropriate sizes of the arrays for your particular problem at hand. However, the following changes need to be made in addition,

1.) The function 'solveBurgers' from the shipping 1D Burger's example should be changed to your particular PDE, namely the 2D Navier-Stokes equation,

2.) The function 'modelGradients' from the shipping 1D Burger's example should be changed to account for the loss based on the 2D Navier-Stokes equation.

Lastly, the current layer architecture with the nine fully connected layers might not be appropriate for your particular problem at hand. In this case you may need to adjust the layer architecture accordingly.

I hope that this information helps.

Kind regards,

Andreas