Physics informed Neural Network

Dear Matlab Community Members,

i would like to apply my equations in physics-informed neural networks, so i reffered solving Burger's equation L-BFGS method. For example i would like to apply 1D equation

d^u/dt=(d^2g/dx^2)-d^4u/dx^4; where g=2*u*(1-u)*(1-2*u) with 
u(x=0,t)=0, u(x=1,t)=0 and 
u(t=0,x)=sin(4*pi*x/L)

Model Loss Function

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
U = model(parameters,X,T);
G=2.*U.*(1-U).*(1-2.*U);
gradientsG = dlgradient(sum(G,"all"),{X,T},EnableHigherDerivatives=true);
Gx = gradientsG{1};
Gxx = dlgradient(sum(Gx,"all"),X,EnableHigherDerivatives=true);
gradientsU = dlgradient(sum(U,"all"),{X,T},EnableHigherDerivatives=true);
Ux = gradientsU{1};
Ut = gradientsU{2};
Uxx = dlgradient(sum(Ux,"all"),X,EnableHigherDerivatives=true);
Uxxx = dlgradient(sum(Uxx,"all"),X,EnableHigherDerivatives=true);
Uxxxx = dlgradient(sum(Uxxx,"all"),X,EnableHigherDerivatives=true);
f=Ut-Gxx+2.*Uxxxx;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

i have made the above changes in the code...but i didnt get any expected results....can u please help me with this...?

[N A POORNA CHANDRA]

hi sankarganesh what are the expected resutls you were expecting from simulating the code

[Sankarganesh P]

@N A POORNA CHANDRA,

i campared this results with finite difference results for the same equations..its not matching with FDM results(totally different)...i have to know, whether the implementation of above equation in L-BFGS is correct or have to do any modifications..?

[Ben]

@Sankarganesh P I notice that your ODE is

but the model loss function uses 2.*Uxxxx. That seems like an issue at first glance.

Otherwise your model loss function looks as expected to me. Do you get good training losses?

Of course a small training loss only means the network is "nearly" solving the ODE at the training points - for complex ODEs/PDEs this may only give weak approximations to the true solution in general.

[Sankarganesh P]

@Ben Thankz for the correction. i will try now... is it possible to use this LBFGS for 2D problems like d^2T/dx^2+d^2T/dy^2=DT/dt..?