Implementing the 1D wave equation

[engsbk]

Thank you for the great contribution. PINNs have been an amazing tool in scientific computing and this enhancement makes it better. I replicated the code of burger's equation and made changes to model the 1D wave equation with a source term:

@tf.function
def f_model(x,t):
    c = tf.constant(1, dtype = tf.float32)
    Amp = tf.constant(1, dtype = tf.float32)
    frequency = tf.constant(1, dtype = tf.float32)
    sigma = tf.constant(0.5, dtype = tf.float32)
    source_x_coord = tf.constant(0, dtype = tf.float32)

    Gaussian_impulse =  Amp * tf.exp(-(1/(sigma**2))*(x-source_x_coord)**2)
    S = Gaussian_impulse * tf.sin( 1 * tf.constant(math.pi)  * frequency * t )
    u = u_model(tf.concat([x,t], 1))
    u_x = tf.gradients(u,x)
    u_xx = tf.gradients(u_x, x)
    u_t = tf.gradients(u,t)
    u_tt = tf.gradients(u_t,t)
    f_u = u_tt + (c**2) * u_xx - S

    return f_u

However, the output is not quite what I expected.

I tried a couple of network architectures and learning rates, but the result seems to be the same. What would you suggest to get more accurate results? Also, I saw how you placed the boundary conditions but I'm not sure how to specify the initial condition? Is it extracted from the exact solution?

Thanks again! Awaiting your feedback.

[levimcclenny]

@engsbk I would recommend trying TensorDiffEq to specify your own IC, etc. It can absolutely be done in the code here, however, by modifying x0 to be an [N0, 1] tensor representing your function of x. I do use the solution to pull the IC but that is not required and should likely be fixed.

The script to replicate this Burgers example in TDQ is here - https://github.com/tensordiffeq/TensorDiffEq/blob/main/examples/burgers-new.py

You would just modify func_ic to be your function of x.

Please let me know if that helps or if we need to discuss further.

Also, sorry for the delayed response, it's been a long week on my side.

[engsbk]

Thank you so much for your reply, @levimcclenny !

I actually tried it with modifications and it works great. I have some follow-ups that I'll post on the tensordiffeq page for consistency.