jdtoscano94 / learning-scientific_machine_learning_residual_based_attention_pinns_deeponets Goto Github PK

On the DeepONet anti-derivative notebook, the last example shows how to approximate the anti-derivative for $u(x)=cos(2\pi x),∀x\in[0,1]$. However, the u_fn lambda is defined as $u(x)=-cos(2\pi x),∀x\in[0,1]$.

From what I could understand $s(x)$ would be equals to $-\frac{1}{2\pi}sin(2\pi x)$ in this case.

Actual text:

Let's obtain the anti-derivative of a trigonometric function. However, remember that this neural operator works for $x\in[0,1]$ when the antiderivative's initial value ($s(0)=0$). To fulfill that conditions, we will use $u(x)=cos(2\pi x),∀x\in[0,1]$.

#u_fn = lambda x, t: np.interp(t, X.flatten(), gp_sample)
u_fn = lambda t, x: -np.cos(2*np.pi*x) # Should be np.cos(2*np.pi*x)
# Input sensor locations and measurements
x = np.linspace(0, 1, m)
u = u_fn(None,x)
# Output sensor locations and measurements
y =random.uniform(key_train, (m,)).sort()

I want to ask if you think you can use PINN to solve this problem.

We will solve a simple ODE system:

$$ {\frac{dV}{dt}}=10- {G_{Na}m^3h(V-50)} - {G_{K}n^4(V+77)} - {G_{L}(V+54.387)}$$

$${\frac{dm}{dt}}=\left(\frac{0.1{(V+40)}}{1-e^\frac{-V-40}{10}}\right)(1-m) - \left(4e^{\frac{-V-65}{18}}\right)m $$

$$\frac{dh}{dt}= {\left(0.07e^{\frac{-V-65}{20}}\right)(1-h)} - \left(\frac{1}{1+e^\frac{-V-35}{10}}\right)h$$

$$\frac{dn}{dt}= {\left(\frac{0.01(V+55)}{1-e^\frac{-V-55}{10}}\right)}(1-n) - \left(0.125e^{\frac{-V-65}{80}}\right)n$$

$$\qquad \text{where} \quad t \in [0,7],$$

May I ask how to represent a system of differential equations?

with the initial conditions
$$V(0) = -65, \quad m(0) = 0.05 , \quad h(0) = 0.6 , \quad n(0) = 0.32 $$

The reference solution is here, where the parameters $G_{na},G_{k},G_{L}$ are gated variables and whose true values are 120, 36, and 0.3, respectivly.

The code below is the data description

data = np.load('g_Na=120_g_K=36_g_L=0.3.npz')
t = data['t']
v = data['v']
m = data['m']
h = data['h']
n = data['n']

Hi Juan,
Thanks for your helpful video. I subscribed!
However, I have a problem here!
the "from jax.experimental import optimizers" does not work for me! it says "cannot import name 'optimizers' from 'jax.experimental'".
Then I have to switch to CPU and also install "!pip install jax[cpu]==0.2.27" to work!
It is confusing for me and as I searched the net for other people.
Could you please let me know how you use GPU in your video? I have to use my CPU and it takes like a year for training!!!

Thank you

I run your simple ode solver using PINN. It seems that the model perform well on the training interval,

But couldn't manage to predict outside the training interval,

I was surprised, as the paper was considered so good (from my theoretical perspective). Isn't it should learn the periodicity of the solution? Do you have tested it and get better performance outside the training interval? Let me know your thought to improve it.

Thanks for your consideration.

A great repository!Thank you very much for your open source, i also really like your videos about this on your Youtube! It is so awesome！

I have a question about deeponet.

$$ \frac{d y}{d x}+y(x)=\int_0^x e^{t-x} y(t) d t $$

i want to use DeepOnet to solve this integro-differential equations, but i have no idea how to implement it. The main problem is that i don't want to approximate integral op-
erators numerically, i just hope to make it with DeepOnet

Is it possible to solve it using DeepOnet only?

Best Regrads!

Hi, thank you so much for your work! I am wondering about the generalization ability of such PINN? I tried with a simple sine function, when training and testing within the range: [0,2pi], both training loss and validation loss are good. However, when I feed the network with a new set of x, ranges from [2pi, 4pi], then the prediction looks bad. Is it because that the network neve sees such number? I feel like it is memorizing the distribution of things it has been trained on, but not generalized to unseen floats?