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[shuheng-liu]

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[almetricmath]

I'm getting this error when I try to run Example 2 from the get started section of the manual - "cannot import name 'Monitor1D' from 'neurodiffeq.monitors". How do I resolve this error?

[shuheng-liu]

It's most likely because you have an old version of neurodiffeq. Can you install the latest version with pip install -U neurodiffeq and see if the problem persists?

[almetricmath]

I fixed some things in my python environment and upgraded neurodiffeq. It worked fine after that! Thanks!

[Godhand594]

Hello, the neurodiffeq is very good to solve the PDE and ODE.
I have seen the official website and I'm interesting in ODE system.
I think that is there any way to solve the ODE+PDE system by using neural network?
also, the relationship between variable in ODE and in PDE are dependent.
for simple example
1.ODE:du/dt= u+T
2.PDE: ∂T/∂t= ∂T^2/∂x

[Godhand594]

Dear @shuheng-liu ,hello, for the ODE+PDE system, although I know how to customize the layer, is anything that should be customized into the network class? for the above equation system.

[shuheng-liu]

Hi there. No, there's nothing else you need to do. Just make sure the network admits an input tensor of shape (N_SAMPLES, 2), and discards the first column (x) of the input tensor. To enforce initial and boundary conditions, we use the classes defined in neurodiffeq.conditions and pass them to Solver(..., conditions=[...]).

[Godhand594]

Thank you for replying. The following is my consideration:
Eq1:ODE: du/dt=u+T
eq2:PDE: ∂T/∂t = u*∂T^2/∂x
1.When specifying the initial conditions for Eq1 and boundary conditions for Eq2, I put the first one in conditions by using IVP, and put the second one in conditions by using IVBP1D.
2.for the network, because of 2 differential equations,
the network = [ FCNN( n_input_units=1, n_output_units=1,hidden_units=(32, 32, 32)),
FCNN(n_input_units=2, n_output_units=2,hidden_units=(32, 32, 32))]
And I think this part is confusing me, you said that view u as a function of both x and t . Should I change something in the network?
3.The solve_system function is for solving ODE systems. Because of the ODE+PDE system, is it suitable to use solve_system or something recommend?

[shuheng-liu]

Hi @Godhand594 ,

1. For the initial/boundary conditions, you can definitely use the IBVP1D for the second network. But for the first one, since we view u as a function of x and t, we should also use an IBVP (with u(x,t=0) a constant function in x) and zero Neumann condition on both u(x=x0,t) and u(x=x1,t)
2. For the network architecture, both of them should have 2 inputs and 1 output. In the network for u, you can manually separate x and t, and only use t for foward pass.
3. I recommend using the Solver2D class instead of the solve_system2d function. The latter is deprecated.

Have a nice day!

[Godhand594]

I got it. I will try it. Thank you very much!

[elboujaoui]

hello @shuheng-liu, I'm trying to solve wave equation, but I didn't find any way to introduce speed condition. I try for example :

conditions = [
IBVP1D(
t_min=0, t_min_val=lambda x: torch.sin(np.pi * x ),
t_min=0, t_min_prime=lambda x: 0,
x_min=0, x_min_val=lambda t: 0,
x_max=L, x_max_val=lambda t: 0
)
]

and I get the following error :

SyntaxError: keyword argument repeated

could you please help to resolve this issue?

[shuheng-liu]

Hi @elboujaoui , you don't need to supply the t_min=0 argument twice.
Can you try:

conditions = [
    IBVP1D(
        t_min=0, t_min_val=lambda x: torch.sin(np.pi * x), t_min_prime=lambda x: 0,
        x_min=0, x_min_val=lambda t: 0,
        x_max=L, x_max_val=lambda t: 0
    )
]

[shuheng-liu]

On a second look, it appears t_min_prime is not supported yet in IBVP1D. While we're still working on this, you might need to do the reparamterization by yourself for now.

[elboujaoui]

Thank you very much for your answer. I hope you will add this situation to the neurodiffeq package.

[yhl48]

Hi @shuheng-liu, I was looking at the harmonic oscillator ODE example, and was wondering how you enforced the amplitude condition, if you did at all? I tried digging into the code and the paper but could not find anything relevant, though I might have missed out some details. Could you please shed some light on that? Thanks in advance!

[shuheng-liu]

Cool. Can you clarify which paper you are referring to? We actually have a couple of papers on related subjects.

[yhl48]

I was referring to this paper https://zenodo.org/record/3674095#.YT4s854zZhE, thanks!

[shuheng-liu]

Thanks, and I assume you are referring to this problem in our documentation. If you look at the second line in cell 8, it creates an IVP (short for Initial Value Problem) which specifies that u(0) = 0 and u'(0)=1. These conditions should guarantee uniqueness of solution. (Let me know if there's any confusion so far).

Now, the way we enforce this condition is to reparameterize the output of neural networks like this:

This guarantees that, whatever output ANN gives, u(0)=0 and u'(0)=1 always hold.

You can checkout the documentation for IVP here and the implementation here.

[shuheng-liu]

As for this method

neurodiffeq does provide a way of adding a custom loss term (kind of like regularization), but this is not what we did in the tutorial. We go with the second method, namely reparamterization, in the tutorial. Let me know if you want to try the first method using neurodiffeq.

[yhl48]

On further thoughts, you are right, by enforcing the conditions on u(0), u'(0), and the second order ODE, the uniqueness of the solution should be guaranteed. I will check out the links, appreciate it.

I tried the second method and it failed to produce the expected results, hence the confusion, but I just found out what went wrong, and it is working now. Thank you, great library!

[oruen321]

Hello @shuheng-liu , thanks for offering the neurodiffeq! It works really well.
I have a question related to the initial values: Until now I have only trained neural networks for one specific initial value. For example in diffusion, I start with a concentration c0 = c(t=0,x). But is it possible to also train for a range of possible initial values using only 1 network?
Meaning that the inital value is also a input of the PINN. so my PINN would get c0 as an input and can calculate the concentration for the next timestep?
like: c_1 = PINN(c_0, timestep, x), where c_0 can be a random value within a range? Thanks!

[shuheng-liu]

Hi @oruen321, here's an example of how to solve what we call a bundle of solutions. This currently works for ODEs only.

Make sure you have the latest version of the library (v0.6.2 or higher)

https://colab.research.google.com/drive/1L9NGeLoDONWmCr_tZkjIJziF3yo1OJHE?usp=sharing

Unfortunately, bundled solutions for PDEs have not yet been implemented. You're welcome to open an issue to request this feature.

[oruen321]

Hi @shuheng-liu , thanks for the answer. But is there any research paper that show that bundled solutions for PDEs are already implemented? Or can be implemented?
Thanks!

[shuheng-liu]

Hi @shuheng-liu , thanks for the answer. But is there any research paper that show that bundled solutions for PDEs are already implemented? Or can be implemented?

Thanks!

Hi there, my biggest concern is that the boundary condition for a PDE can be a function of time in general. Similarly, the initial condition can be a function of spatial location. I haven't done a thorough literature review, but I struggle to think of a way that allows functions, rather than values, to be passed to the neural network as an input.

[oruen321]

Thanks for the answer. I first started to use the normal example from the documentation "1-D Heat-Equation" I tried to use the dirchlet conditions with u(0,t) = 0.1 and u(L,t) = 2 and the initial value of u(x,0)=0.1. If correctly, this should give a linear tend from 0.1 to 2 on the x axis, or at least at x=L, the value should be 2. but the value on both ends remain 0.1. I changed the conditions to:
t_min=0, t_min_val=lambda x: 0.1,
x_min=0, x_min_val=lambda t: 0.1,
x_max=L, x_max_val=lambda t: 2
with other values staying the same. What is my mistake here?
Thanks!

[aryx6984]

Is it possible to use this library for solving CFD equations?(Navier-stocks and heat transfer equations as a system of PDE)

[shuheng-liu]

Hi there, for heat equations, I think neurodiffeq will do very well. As for Navier-Stokes, you can definitely use neurodiffeq to solve it, but the resulting solution might not be as good as what you can get from traditional methods, say, the finite element method. This is uncharted territory which means there are more research explorations than guaranteed gains.

[aryx6984]

Can I also solve equations with boundary conditions like (u(9,t)=sin(x))?