Batch inconsistency in PINNs!!

Hi there!

I am working on this example: Solve PDE Using Physics-Informed Neural Network

Based on the following highlights extracted from this example:

X0 = [x0IC x0BC1 x0BC2]; %numBoundaryConditionPoints = [25 25];
T0 = [t0IC t0BC1 t0BC2]; %numInitialConditionPoints  = 50;

points = rand(numInternalCollocationPoints,2); %numInternalCollocationPoints=10000
dataX = 2*points(:,1)-1;
dataT = points(:,2);

These mean:

• dataX and dataT are column vectors of length 10000.
• X0, and T0 are two row vectors of length 100, i.e. 50+25+25

The following dlarray formats make sense because of column vectors X, T and row vectors X0, T0.

X = dlarray(dataX,"BC");
T = dlarray(dataT,"BC");
X0 = dlarray(X0,"CB");
T0 = dlarray(T0,"CB");

What I cannot underestant from model function is these parts:

XT = cat(1,X,T);   %====>???!!!
U = forward(net,XT);

and

XT0 = cat(1,X0,T0);
U0Pred = forward(net,XT0);

Shouldn't it be as below?

XT = cat(2,X,T);
U = forward(net,XT);

How it has been worked?

Thanks in advance for any clarification!

Answer 1

[praguna manvi]

Hi @MAHSA YOUSEFI,

As I see, you are looking for an explanation of the function “cat(1, X, T)” which concatenates on dimension 1 for the following Physics-Informed Neural Network (PINN) example:

openExample('nnet/TrainPhysicsInformedNeuralNetworkWithLBFGSAndAutoDiffExample’)

In this example, the network expects input features of size “2C x 1B”, meaning 2 features with a batch size of “B”. You can visualize the network using:

analyzeNetwork(net)

In the example, the dimensions of “X”, “T”, “X0”, and “T0” are in the data format “1(C) x 10000(B) and 1(C) x 100(B)” respectively. Using “cat(1, X, T)”, you create an input feature of size “2(C) x 10000(B)”, which matches the feature input size expected by the neural network. On the other hand, “cat(2, X, T)” would create a size of “1(C) x 20000(B)” on dimension two.

For more information on data format and functions on “dlarray” refer to the following link:

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

[MAHSA YOUSEFI]

Dear Praguna, thanks a lot!

I fully underestand the structure of the network and the requirements for the inputs, and agree with what you mentioned with X0 and Y0.

I will be thankful if you could make it clear why you conclude that X and T are arrays with C×B format, while it was writen that they are B×C.

Please notice to the definition of dataX and dataT. They are column vectors. How it would be possible that we concatanate them with dim=1?

Accorrding to the format B×C, the should concatenate with dim=2 to be suited with input layer.

[praguna manvi]

@MAHSA YOUSEFI, the dimensions of "dataX" and "dataY" in the following lines of the example:

X = dlarray(dataX, "BC");
T = dlarray(dataT, "BC");

are 10000 x 1, which is read in the format as "BC". This results in "X" & "T" being "1(C) x 10000(B)". Therefore, we can concatenate them along dimension 1.

[MAHSA YOUSEFI]

@praguna manvi I can see now!! The key point was then in "the software automatically permutes the dimensions such that the format has dimensions in the order: "S", ''C", "B", and .... ". I appreciate for your answer.