Issue with IC when simulating the free vibration of a simply supported euler beam

[DavidBraun97]

Dear Mr. @lululxvi ,

First of all I would like to thank you for sharing deepxde with the community. Its a wonderful toolbox for Physics Informed ML.

I am currently trying to simulate the free vibration (q(x) = 0) of a simply supported euler beam. That is the problem is as follows: with the IC: In my particular choice I have chosen the IC to be:

So far the output after 25000 epochs of adam with lr=1e-3 and subsequent BFGS optimization the results is as follows: It appears that the general structure of the solution is learned correctly. However, when taking a closer look the magnitude of the NN output appears to be off. See for instance the predicted deflection at t=0, i.e. at IC: Considering above result, the model does not correctly learn the IC. In fact there seems to be an issue of scaling. Above results are the outcome of the following training routine: ####################################### model.compile("adam", lr=1e-3, loss_weights=[1e-3,1,1,500,100]) losshistory, train_state = model.train(epochs=25000) ..... model.compile("L-BFGS-B", loss_weights=[1e-3,1,1,500,100]) losshistory, train_state = model.train() #######################################

So far I do not know where the cause for this error in magnitude lies in. I think that I implemented the problem correctly. For illustration, please find the problem definition below:

####################################### from future import absolute_import from future import division from future import print_function import numpy as np import deepxde as dde from deepxde.backend import tf import matplotlib.pyplot as plt from scipy.interpolate import griddata from plotly.subplots import make_subplots import plotly.graph_objects as go

def dydt(x,y): dy_dt = tf.gradients(y, x)[0][:, 1:2] return dy_dt def dydxx(x,y): dy_dx = tf.gradients(y, x)[0][:, :1] dy_xx = tf.gradients(dy_dx, x)[0][:, :1] return dy_xx

def pde(x, y): dy_xt = tf.gradients(y, x)[0] dy_x,dy_t = dy_xt[:, :1], dy_xt[:, 1:]

dy_xx = tf.gradients(dy_x, x)[0][:, :1] dy_tt = tf.gradients(dy_t, x)[0][:, 1:]

dy_xxx = tf.gradients(dy_xx, x)[0][:, :1] dy_xxxx = tf.gradients(dy_xxx, x)[0][:, :1]

return dy_xxxx + dy_tt

def boundaryinitial(x, ): return np.isclose(x[-1], 0)

def func(x): x, t = x[:, :1], x[:, 1:] return np.sin(3 np.pi x) np.cos(9(np.pi*2)t)

geom = dde.geometry.Interval(0, 1) timedomain = dde.geometry.TimeDomain(0, 0.01) geomtime = dde.geometry.GeometryXTime(geom, timedomain)

bcs = [ dde.DirichletBC(geomtime, lambda x: 0,lambda _, on_boundary: onboundary),
dde.OperatorBC(geomtime, lambda x, y, : dydxx(x,y), lambda _, on_boundary: on_boundary)
]

ics = [ dde.IC(geomtime, lambda x: np.sin(3 np.pi x), lambda _, on_initial: oninitial), dde.OperatorBC(geomtime, lambda x,y,: dydt(x,y), lambda _, on_initial: on_initial) ]

data = dde.data.TimePDE( geomtime, pde, bcs + ics, num_domain=3000, num_boundary=600, num_initial=900, solution=func, num_test=100, train_distribution="uniform", ) layer_size = [2] + [100]*3+ [1] last activation = "tanh" initializer = "Glorot uniform" net = dde.maps.FNN(layer_size, activation,initializer)

model = dde.Model(data, net) #######################################

It would be very kind of you if you (or anybody else out there) could give me some feedback on my implementation. Maybe someone has an idea where things go wrong. Besides that, I am not 100% sure if my PDE,BC and IC are correctly defined.

I am very thankful for any feedback! Thanks again for your great contribution @lululxvi.

Cheers, David

[lululxvi]

IC should be

dde.IC(geomtime, lambda x: np.sin(3* np.pi * x[:, 0:1]), lambda _, on_initial: on_initial)

x is (x, t).

[DavidBraun97]

Thank you very much @lululxvi ! This helped a lot.

I am closing the issue now. Cheers.