Parametric Burgers + hard Dirichlet BC doubt

[GuillemBarroso]

Dear @lululxvi,

First, thank you for your dedicated work. I have to questions regarding DeepXDE.

1. Parametric Burgers

I am interested in parametric forward problems and I have started with the 1D Burgers example available in the examples section. As a first test to introduce a parameter dependency on this test problem, I have modified the initial condition of the problem as

u(x,0) = sin(mupix), with mu \in [1,3]

so that I can use different values of mu to modify the problem's behaviour. I can successfully train several NN that take x and t as inputs and predicts u with a good accuracy for different values of mu. Note that the boundary conditions u(0,t) = u(1, t) = 0 are only satisfied for integer values of mu. That is why I have modified the right BC to also change with time, see the modified code attached.

Now, in the next stage I would like to include mu as a parametric direction to the problem. That is, a NN that takes mu, x and t and predicts u(mu, x, t). And here is where I am not too sure how to proceed. I have seen 2D and 3D geometries in time-dependent problems in the examples provided, but it is not clear to me how to add a "parametric direction". Again, my goal is to define a NN with 3 inputs mu, x and t and the training data stored in "x" should have all the points' locations in mu, x and t (as opposed to now that I only have x and t).

I hope I made myself clear. Please let me know otherwise.

2. Enforcing Dirichlet BC strongly

As discussed in the issues and as stated in https://arxiv.org/abs/1907.04502, DeepXDE enforces the BC softly. For the parametric Burgers mentioned in question 1) I believe that enforcing strongly the Dirichlet BCs would ease the training (that is the idea I got reading the issues). However, I am not fully understanding the methodology followed when using net.apply_output_transform.

In the paper you mention that for u(0) = u(1) with \Omega = [0, 1], one could simply choose the surrogate model u_new(x) = x(x-1)N(x). My question is, is that transformation (using net.apply_output_transform) going to be applied to all outputs regardless of the value of x? Of course, for x = 0 and x = 1, u_new(x) is going to be set equal to 0. But for x = 0.5 it would be u_new(x) = -0.25*N(x), which would modify the solution in the interior of the domain. Am I missing something? Also, looking at how I define the right BC (at x=1), please see the code attached, do you think it is possible to use a similar method to strongly impose these Dirichlet BC?

Thank you very much for your time. I greatly appreciate your help.

Best regards,

Guillem
paramBurguers_sin.py.zip

[GuillemBarroso]

Please find attached the code in which I tried to use a 2D geometry (x, mu) in order to construct the neural network for the parametric Burgers problem.

I am getting a ValueError when compiling the model in line 116

paramBurguers.py.zip

[lululxvi]

To include the parameter mu, you can simply assume mu is another space coordinate. In DeepXDE, there is no difference between x and mu. So, if we change the notation from mu to y, then it is u(x, y, t), where u is defined in a 2D space.

For the hard constraints, you can also check this paper https://arxiv.org/abs/2102.04626 for more details.

[rodrigogdourado]

@GuillemBarroso did you manage to solve the parametric problem?

[GuillemBarroso]

My apologies for not following up on this issue.

@GuillemBarroso did you manage to solve the parametric problem?

Yes! I did manage to get the parametric problem running @rodrigogdourado. As pointed out by @lululxvi, it was just a matter of considering it as an extra space coordinate, so you would be solving a 3D (x, mu, t) problem.

That was a year ago so I will not be able to be more detailed. I just remember that I had a bug in my code (see attachment file in the original comment) in the pde definition. In line 71, the derivative with respect to time dy_t should now be computed as dde.grad.jacobian(y, x, i=0, j=2). Note the 2 instead of 1 (time is now the 3rd component instead of the 2nd).

Good luck!

[rodrigogdourado]

Thank you very much @GuillemBarroso! It worked very well.

[rodrigofarias-MECH]

My apologies for not following up on this issue.

@GuillemBarroso did you manage to solve the parametric problem?

Yes! I did manage to get the parametric problem running @rodrigogdourado. As pointed out by @lululxvi, it was just a matter of considering it as an extra space coordinate, so you would be solving a 3D (x, mu, t) problem.

That was a year ago so I will not be able to be more detailed. I just remember that I had a bug in my code (see attachment file in the original comment) in the pde definition. In line 71, the derivative with respect to time dy_t should now be computed as dde.grad.jacobian(y, x, i=0, j=2). Note the 2 instead of 1 (time is now the 3rd component instead of the 2nd).

Good luck!

Hi @GuillemBarroso . Since you are ahead of me, could you explain the i and j in this operator dde.grad.jacobian(y, x, i=0, j=2)?

I'm still making some confusions with this numbering system. Which are the numbers for the different derivatives, like dy_dx, dy_dt, dy_dxx and etc?

[rodrigogdourado]

@rodrigofarias-MECH are you from brazil?

If your problem involves 2-dimensional coordinates (x1,x2), is time-dependent, and requires solving for only one variable (such as in a 2D heat conduction problem), then

dy_dx1 -> i=0,j=0
dy_dx2 -> i=0,j=1
dy_dt -> i=0,j = 2

now, consider a 2D navier-stokes equation. In this case you are solving the problem for u, v, p, so your y matrix is composed by 3 columns. Thus,

u = y[:, 0:1]
v = y[:, 1:2]
p = y[:, 2:3]

du_dx1 - > i=0,j=0
du_dx2 - > i=0,j=1
du_dt - > i=0,j=2
dv_dx1 -> i=1, j=0
dv_dx2 -> i=1, j=1
dv_dt -> i=1, j=2
dp_dx1 -> i=2, j=0
dp_dx2 -> i=2, j=1

so, j is related to the input and i is related to the output parameters

[rodrigofarias-MECH]

@rodrigofarias-MECH are you from brazil?

If your problem involves 2-dimensional coordinates (x1,x2), is time-dependent, and requires solving for only one variable (such as in a 2D heat conduction problem), then

dy_dx1 -> i=0,j=0 dy_dx2 -> i=0,j=1 dy_dt -> i=0,j = 2

now, consider a 2D navier-stokes equation. In this case you are solving the problem for u, v, p, so your y matrix is composed by 3 columns. Thus,

u = y[:, 0:1] v = y[:, 1:2] p = y[:, 2:3]

du_dx1 - > i=0,j=0 du_dx2 - > i=0,j=1 du_dt - > i=0,j=2 dv_dx1 -> i=1, j=0 dv_dx2 -> i=1, j=1 dv_dt -> i=1, j=2 dp_dx1 -> i=2, j=0 dp_dx2 -> i=2, j=1

so, j is related to the input and i is related to the output parameters

Yes I'm from RJ-Brazil!

Thank you, know I understood the indexes! Good example this for Navier-stokes.

[rodrigogdourado]

@rodrigofarias-MECH

I work at ITA. You are the first brazilian I know interested in PINN. If you would like to share your work with me, maybe we can work together in something. You can text me on researchgate

https://www.researchgate.net/profile/Rodrigo-Dourado-2?ev=hdr_xprf

[rodrigofarias-MECH]

@rodrigofarias-MECH

I work at ITA. You are the first brazilian I know interested in PINN. If you would like to share your work with me, maybe we can work together in something. You can text me on researchgate

https://www.researchgate.net/profile/Rodrigo-Dourado-2?ev=hdr_xprf

I work at LNTSOLD-COPPE/UFRJ. I send you a message in researchgate and linkedin.
Regards.