Parameter Identification with PINN (Physics Informed NN) | Application Package Repository Telkom University (2024)

Hello,
I have a system of PDEs simulating a heat transfer system, and I have experimental data of the behaviour of the system. I am trying to use a PINN for the identification of 4 unkwnon parameters.
I am trying first to validate the model with known parameters, but I am unable to reach any reasonable parameter value. The learning rate is too slow, any recommendations?
Thanks for the help!
To run the code you will need to download the following .mat: https://cyclomed-my.sharepoint.com/:u:/g/personal/administracion_cyclomed_onmicrosoft_com/ETlg6E6520JDnBoIOv35Bq0B0o6ZCXvAvvqG3Gn1Ht3ssA?e=hKclKH
For context this is a simplified version of the PDE:

% Identify parameters of single blow equation
% 1) Fluid (u1) –> B*du1/dt + du1/dx = 1/Pe*d2u1/dx^2 + NTU*(u2 – u1) + NTU_W(u3 – u1)
% 2) Regenerator (u2) –>du2/dx = K_R*d2u2/dx^2 – NTU*(u2 – u1)
% 3) Wall (u3) –> du3/dx = R_TC*K_W*d2u3/dx^2 – R_TC*NTU_W*(u3 – u1) – NTU_W(u3 – u3_init)

% Load Experiment data
load("Test 11.mat");

%% Neural Network Definition
batchSize = 500;
dimension = 2; % X = (t,x)

% Set up neural net.
hiddenSize = 100;
net = [
featureInputLayer(dimension)
fullyConnectedLayer(hiddenSize)
tanhLayer
fullyConnectedLayer(hiddenSize)
tanhLayer
fullyConnectedLayer(hiddenSize)
tanhLayer
fullyConnectedLayer(3)];

% Create struct of parameters
parameters.net = dlnetwork(net);

% Unknown PDE parameters
parameters.Pe = dlarray(1000);
parameters.NTU = dlarray(200);
parameters.NTU_W = dlarray(1);
parameters.NTU_EXT = dlarray(50);

% Known PDE parameters
constants.B = B;
constants.K_R = K_R;
constants.K_W = K_W;
constants.R_TC = R_TC;
constants.R_Q = R_Q;
constants.T_W = u3(1,1);

%% Experiment data — Real PDE data

% Define the size of the original matrix
[rows, cols] = size(u1);

% Generate the grid for the original matrix
[X, Y] = meshgrid(1:cols, 1:rows);

% Generate the grid for the interpolated matrix
[XI, YI] = meshgrid(linspace(1, cols, batchSize), linspace(1, rows, batchSize));

% Interpolate the matrix along each dimension
UTest1 = (interp2(X, Y, u1, XI, YI));
UTest2 = (interp2(X, Y, u2, XI, YI));
UTest3 = (interp2(X, Y, u3, XI, YI));

% Select random points
[rows, columns] = size(UTest1);
X = ceil(rand(batchSize,1)*columns);
Y = ceil(rand(batchSize,1)*rows);

for k = 1 : length(X)
row = Y(k);
col = X(k);
UTest.u1(k) = UTest1(row, col);
UTest.u2(k) = UTest2(row, col);
UTest.u3(k) = UTest3(row, col);
end

% Transfor to dlarray
x = dlarray((X./batchSize)’, "CB");
t = dlarray((Y.*t(end)./batchSize)’, "CB");
X = [t;x];

% Experiment data
UTest.u1 = dlarray(UTest.u1, "CB");
UTest.u2 = dlarray(UTest.u2, "CB");
UTest.u3 = dlarray(UTest.u3, "CB");

%% Solve inverse PINN
% Training loop
avgG = [];
avgSqG = [];
maxIters = 10000000;
lossFcn = dlaccelerate(@modelLoss);

for iter = 1:maxIters
[loss,gradients] = dlfeval(lossFcn,X,parameters,constants,UTest);

[parameters,avgG,avgSqG] = adamupdate(parameters,gradients,avgG,avgSqG,iter);

if mod(iter, 1000) == 0
fprintf("Iteration : %d, Loss : %.4f n",iter, extractdata(loss));
fprintf("Iteration: %d, Predicted Pe: %.3f, Actual Pe: %.3f, Predicted NTU: %.3f, Actual NTU: %.3fn", …
iter, extractdata(parameters.Pe), Pe, extractdata(parameters.NTU), NTU);
fprintf("Iteration: %d, Predicted NTU_W: %.3f, Actual NTU_W: %.3f, Predicted NTU_EXT: %.3f, Actual NTU_EXT: %.3fn", …
iter, extractdata(parameters.NTU_W), NTU_W, extractdata(parameters.NTU_EXT), NTU_Q);
end
end

%% PINN definition
function [loss,gradients] = modelLoss(X,parameters,constants,UTest)
% X = (t,x).
% PDE
u = predict(parameters.net, X);
u1 = u(1,:);
u2 = u(2,:);
u3 = u(3,:);

% First partial derivatives
du1 = dlgradient(sum(u1,2), X, RetainData=true, EnableHigherDerivatives=true);
du2 = dlgradient(sum(u2,2), X, RetainData=true, EnableHigherDerivatives=true);
du3 = dlgradient(sum(u3,2), X, RetainData=true, EnableHigherDerivatives=true);

du1dt = du1(1,:);
du1dx = du1(2,:);
du2dt = du2(1,:);
du2dx = du2(2,:);
du3dt = du3(1,:);
du3dx = du3(2,:);

% Second partial derivatives
d2u1 = dlgradient(sum(du1dx,2),X,RetainData=true);
d2u2 = dlgradient(sum(du2dx,2),X,RetainData=true);
d2u3 = dlgradient(sum(du3dx,2),X,RetainData=true);

d2u1dx2 = d2u1(2,:);
d2u2dx2 = d2u2(2,:);
d2u3dx2 = d2u3(2,:);

% PDE residual
odeResidual = (constants.B*du1dt + du1dx – 1/parameters.Pe*d2u1dx2 – parameters.NTU*(u2 – u1) – parameters.NTU_W*(u3 – u1)).^2;
odeResidual = odeResidual + (du2dt – constants.K_R*d2u2dx2 + parameters.NTU*(u2 – u1)).^2;
odeResidual = odeResidual + (du3dt – constants.K_W*d2u3dx2 + constants.R_TC*parameters.NTU_W*(u3 – u1) + constants.R_Q*parameters.NTU_EXT*(u3 – constants.T_W)).^2;

% Compute the mean square error of the ODE residual
pdeLoss = mean(odeResidual,"all");

% Compute the L2 difference between the predicted xpred and the true x.
reconstructionLoss = l2loss(u1,UTest.u1) + l2loss(u2,UTest.u2) + l2loss(u3,UTest.u3);

loss = pdeLoss + reconstructionLoss;
[gradients.net, gradients.Pe, gradients.NTU, gradients.NTU_W, gradients.NTU_EXT] = dlgradient(pdeLoss,parameters.net.Learnables,parameters.Pe, parameters.NTU, parameters.NTU_W, parameters.NTU_EXT);
endHello,
I have a system of PDEs simulating a heat transfer system, and I have experimental data of the behaviour of the system. I am trying to use a PINN for the identification of 4 unkwnon parameters.
I am trying first to validate the model with known parameters, but I am unable to reach any reasonable parameter value. The learning rate is too slow, any recommendations?
Thanks for the help!
To run the code you will need to download the following .mat: https://cyclomed-my.sharepoint.com/:u:/g/personal/administracion_cyclomed_onmicrosoft_com/ETlg6E6520JDnBoIOv35Bq0B0o6ZCXvAvvqG3Gn1Ht3ssA?e=hKclKH
For context this is a simplified version of the PDE: