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.
