How can I apply the second order derivative to boundary condition in loss function of PINN-based deep learning network ?

Question

Mansung Kang 2023-8-15

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2008692-how-can-i-apply-the-second-order-derivative-to-boundary-condition-in-loss-function-of-pinn-based-dee

编辑： Ruchika 2023-8-16

采纳的回答： Ruchika

在 MATLAB Online 中打开

I want to apply the second order derivate to boundary condition in loss function of PINN-based deep learning network.

I am trying to run the below MATLAB example code.

openExample('nnet/TrainPhysicsInformedNeuralNetworkWithLBFGSAndAutoDiffExample')

In the example code, there are two loss function consisted of Burger's equation and initial, boundary conditions.

Burger's equation: ∂u/∂t+u*u∂/x∂−0.01/π*(∂^2u/∂x^2)=0

initial and boundary conditions: u(x,t=0)=−sin(πx) as the initial condition, and u(x=−1,t)=0 and u(x=1,t)=0

numBoundaryConditionPoints = [25 25];
x0BC1 = -1*ones(1,numBoundaryConditionPoints(1));
x0BC2 = ones(1,numBoundaryConditionPoints(2));
t0BC1 = linspace(0,1,numBoundaryConditionPoints(1));
t0BC2 = linspace(0,1,numBoundaryConditionPoints(2));
u0BC1 = zeros(1,numBoundaryConditionPoints(1));
u0BC2 = zeros(1,numBoundaryConditionPoints(2));
numInitialConditionPoints  = 50;
x0IC = linspace(-1,1,numInitialConditionPoints);
t0IC = zeros(1,numInitialConditionPoints);
u0IC = -sin(pi*x0IC);
X0 = [x0IC x0BC1 x0BC2];
T0 = [t0IC t0BC1 t0BC2];
U0 = [u0IC u0BC1 u0BC2];
function [loss, gradients] = modelLoss(net, X, T, X0, T0, U0)
% Make predictions with the initial conditions.
XT = cat(1, X, T);
U = forward(net, XT);
% Calculate derivatives with respect to X and T.
gradientsU = dlgradient(sum(U, "all"), {X, T}, EnableHigherDerivatives = true);
Ux = gradientsU{1};
Ut = gradientsU{2};
% Calculate second-order derivates with respect to X.
Uxx = dlgradient(sum(Ux, "all"), X, EnableHigherDerivatives = true);
% Calculate mseF. Enforce Burger's equation.
f = Ut + U.*Ux - (0.01./pi).*Uxx;
zeroTarget = zeros(size(f), "like", f);
mseF = l2loss(f, zeroTarget);
% Calculate mseU. Enforce initial and boundary conditions.
XT0 = cat(1, X0, T0);
U0Pred = forward(net, XT0);
mseU = l2loss(U0Pred, U0);
% Calculated loss to be minimized by combining errors.
loss = mseF + mseU;
% Calculate gradients with respect to the learnable parameters.
gradients = dlgradient(loss, net.Learnables);
end

I want to change the boundary condition to second order derivate formed as u''(x=−1,t)=0 and u''(x=1,t)=0.

How can I apply the second order derivate boundary conditions (mseU)?

Could you give me the solution ?

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Answer 1

Ruchika 2023-8-16

编辑：Ruchika 2023-8-16

在 MATLAB Online 中打开

Hi, to apply the second-order derivative boundary conditions in the loss function, you need to modify the modelLoss function in the MATLAB example code. Here's how you can modify it to enforce the second-order derivative boundary conditions u''(x=-1,t) = 0 and u''(x=1,t) = 0:

function [loss, gradients] = modelLoss(net, X, T, X0, T0, U0)
    % Make predictions with the initial conditions.
    XT = cat(1, X, T);
    U = forward(net, XT);
    % Calculate derivatives with respect to X and T.
    gradientsU = dlgradient(sum(U, "all"), {X, T}, EnableHigherDerivatives = true);
    Ux = gradientsU{1};
    Ut = gradientsU{2};
    % Calculate second-order derivatives with respect to X.
    Uxx = dlgradient(sum(Ux, "all"), X, EnableHigherDerivatives = true);
    % Calculate mseF. Enforce Burger's equation.
    f = Ut + U.*Ux - (0.01./pi).*Uxx;
    zeroTarget = zeros(size(f), "like", f);
    mseF = l2loss(f, zeroTarget);
    % Calculate mseU. Enforce initial and boundary conditions.
    XT0 = cat(1, X0, T0);
    U0Pred = forward(net, XT0);
    Uxx0 = dlgradient(sum(U0Pred, "all"), X0, EnableHigherDerivatives = true);  % Calculate second-order derivatives at boundary points
    % Modify mseU to enforce second-order derivative boundary conditions
    mseU = l2loss(U0Pred, U0) + l2loss(Uxx0(1:numBoundaryConditionPoints(1)), zeros(1,numBoundaryConditionPoints(1))) + l2loss(Uxx0(numBoundaryConditionPoints(1)+1:end), zeros(1,numBoundaryConditionPoints(2)));
    % Calculate loss to be minimized by combining errors.
    loss = mseF + mseU;
    % Calculate gradients with respect to the learnable parameters.
    gradients = dlgradient(loss, net.Learnables);
end

In the modified code, we calculate the second-order derivatives at the boundary points using ’dlgradient’ and store them in Uxx0. Then, we modify ‘mseU’to include the loss term for the second-order derivative boundary conditions by using l2loss to compare Uxx0 with zeros. You need to update the ‘numBoundaryConditionPoints’variable to match the number of boundary condition points you want to use for the second-order derivative boundary conditions.