Inverse problem: Euler-Bernoulli Beam

[tsarikahin]

Dear LuLu,

I think working with higher derivatives is a bit trickier.

I wanted to invert the problem of Euler as given in forward problems beam.

"""Backend supported: tensorflow.compat.v1, tensorflow, pytorch"""
import deepxde as dde
import numpy as np
import matplotlib.pyplot as plt

def ddy(x, y):
    return dde.grad.hessian(y, x, component=0, i=0, j=0)


def dddy(x, y):
    return dde.grad.jacobian(ddy(x, y), x)


def pde(x, y):
    q = y[:,1:2]
    dy_xx = ddy(x, y)
    dy_xxx = dde.grad.jacobian(dy_xx, x)
    dy_xxxx = dde.grad.jacobian(dy_xxx, x)
    #dy_xxxx = dde.grad.hessian(dy_xx, x, component=0, i=0, j=0) This does not work since dy_xx has one dimension, not two
    return dy_xxxx + q #--> it was set as 1


def boundary_l(x, on_boundary):
    return on_boundary and np.isclose(x[0], 0)


def boundary_r(x, on_boundary):
    return on_boundary and np.isclose(x[0], 1)


def func(x):
    return -(x ** 4) / 24 + x ** 3 / 6 - x ** 2 / 4 #solution for displacement

def func_q(x):
    return np.ones(x.shape[0])


geom = dde.geometry.Interval(0, 1)

bc1 = dde.icbc.DirichletBC(geom, lambda x: 0, boundary_l)
bc2 = dde.icbc.NeumannBC(geom, lambda x: 0, boundary_l)
bc3 = dde.icbc.OperatorBC(geom, lambda x, y, _: ddy(x, y), boundary_r)
bc4 = dde.icbc.OperatorBC(geom, lambda x, y, _: dddy(x, y), boundary_r)

def gen_traindata(num):
    # generate num equally-spaced points from -1 to 1
    xvals = np.linspace(0, 1, num).reshape(num, 1)
    uvals = -(xvals ** 4) / 24 + xvals ** 3 / 6 - xvals ** 2 / 4
    return xvals, uvals

ob_x, ob_u = gen_traindata(100)
observe_u = dde.icbc.PointSetBC(ob_x, ob_u, component=0)

data = dde.data.PDE(
    geom,
    pde,
    [bc1, bc2, bc3, bc4, observe_u],
    num_domain=100,
    num_boundary=2,
    num_test=100,
    anchors=ob_x
)

layer_size = [1, [30, 30], [30, 30], [30, 30], 2]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.nn.PFNN(layer_size, activation, initializer)

model = dde.Model(data, net)

model.compile("adam", lr=0.001)
losshistory, train_state = model.train(epochs=5000, display_every=100)

model.compile("L-BFGS-B")
losshistory, train_state = model.train(display_every=100)

X_domain = geom.random_points(100)
X_boun = geom.random_boundary_points(2)
X = np.vstack((X_domain,X_boun))
predictions = model.predict(X)
displacement = predictions[:,0]
force = predictions[:,1]

# displacement
fig, ax = plt.subplots()
ax.scatter(X,displacement,label="prediction")
ax.scatter(X,func(X),label="true")
ax.legend()
plt.savefig("disp.png")
plt.show()

# force
fig, ax = plt.subplots()
ax.scatter(X,force,label="prediction")
ax.scatter(X,func_q(X),label="true")
ax.legend()
plt.savefig("force.png")
plt.show()

In the forward problem, q is given as 1, so I want to find q(x)=1, not as a scalar but as a function, it can be any function actually. For simplicity, let's keep it as is. So q=1 and the solution is given.

I constructed the network and solve the inverse problem. I am kinda sure that it is correctly implemented.

Displacement field

Force field

It is obvious that desired solution is reached but why is the force field not correct?

When I use external_trainable_variable method to identify the scalar q, it works. But I want to get the vector field as well in case finding a complex parameter is desired.

I saw Poisson inverse example and I used the same idea. I don't get why it does not work

Thanks for your help.

Here is also loss values

Step      Train loss                                                      Test loss                                                       Test metric
0         [6.39e-02, 0.00e+00, 9.88e-02, 1.10e-02, 4.05e-06, 1.22e-02]    [6.35e-02, 0.00e+00, 9.88e-02, 1.10e-02, 4.05e-06, 1.22e-02]    []  
100       [2.75e-06, 4.02e-04, 3.66e-04, 1.17e-05, 3.71e-06, 1.49e-03]    [2.68e-06, 4.02e-04, 3.66e-04, 1.17e-05, 3.71e-06, 1.49e-03]    []  
200       [5.74e-06, 3.82e-04, 3.59e-04, 1.36e-05, 1.62e-05, 1.45e-03]    [6.50e-06, 3.82e-04, 3.59e-04, 1.36e-05, 1.62e-05, 1.45e-03]    []  
300       [1.55e-05, 3.74e-04, 3.51e-04, 1.91e-05, 2.38e-05, 1.42e-03]    [1.79e-05, 3.74e-04, 3.51e-04, 1.91e-05, 2.38e-05, 1.42e-03]    []  
400       [2.26e-05, 3.69e-04, 3.44e-04, 2.20e-05, 2.77e-05, 1.39e-03]    [2.64e-05, 3.69e-04, 3.44e-04, 2.20e-05, 2.77e-05, 1.39e-03]    []  
500       [2.62e-05, 3.63e-04, 3.39e-04, 2.27e-05, 2.86e-05, 1.37e-03]    [3.11e-05, 3.63e-04, 3.39e-04, 2.27e-05, 2.86e-05, 1.37e-03]    []  
600       [2.83e-05, 3.58e-04, 3.34e-04, 2.23e-05, 2.79e-05, 1.35e-03]    [3.39e-05, 3.58e-04, 3.34e-04, 2.23e-05, 2.79e-05, 1.35e-03]    []  
700       [3.04e-05, 3.51e-04, 3.29e-04, 2.15e-05, 2.65e-05, 1.32e-03]    [3.65e-05, 3.51e-04, 3.29e-04, 2.15e-05, 2.65e-05, 1.32e-03]    []  
800       [3.32e-05, 3.44e-04, 3.22e-04, 2.04e-05, 2.47e-05, 1.30e-03]    [3.98e-05, 3.44e-04, 3.22e-04, 2.04e-05, 2.47e-05, 1.30e-03]    []  
900       [3.73e-05, 3.35e-04, 3.14e-04, 1.91e-05, 2.26e-05, 1.26e-03]    [4.43e-05, 3.35e-04, 3.14e-04, 1.91e-05, 2.26e-05, 1.26e-03]    []  
1000      [4.31e-05, 3.24e-04, 3.04e-04, 1.76e-05, 2.02e-05, 1.22e-03]    [5.03e-05, 3.24e-04, 3.04e-04, 1.76e-05, 2.02e-05, 1.22e-03]    []  
1100      [5.09e-05, 3.09e-04, 2.91e-04, 1.60e-05, 1.77e-05, 1.17e-03]    [5.80e-05, 3.09e-04, 2.91e-04, 1.60e-05, 1.77e-05, 1.17e-03]    []  
1200      [6.17e-05, 2.90e-04, 2.73e-04, 1.45e-05, 1.57e-05, 1.09e-03]    [6.86e-05, 2.90e-04, 2.73e-04, 1.45e-05, 1.57e-05, 1.09e-03]    []  
1300      [7.34e-05, 2.64e-04, 2.48e-04, 1.33e-05, 1.45e-05, 9.95e-04]    [8.00e-05, 2.64e-04, 2.48e-04, 1.33e-05, 1.45e-05, 9.95e-04]    []  
1400      [6.97e-05, 2.28e-04, 2.14e-04, 1.19e-05, 1.39e-05, 8.58e-04]    [7.58e-05, 2.28e-04, 2.14e-04, 1.19e-05, 1.39e-05, 8.58e-04]    []  
1500      [4.59e-05, 1.76e-04, 1.63e-04, 8.50e-06, 1.21e-05, 6.65e-04]    [5.06e-05, 1.76e-04, 1.63e-04, 8.50e-06, 1.21e-05, 6.65e-04]    []  
1600      [7.82e-05, 1.39e-04, 1.31e-04, 7.27e-06, 9.16e-06, 5.15e-04]    [8.10e-05, 1.39e-04, 1.31e-04, 7.27e-06, 9.16e-06, 5.15e-04]    []  
1700      [9.64e-05, 1.11e-04, 9.61e-05, 8.64e-06, 9.22e-06, 4.37e-04]    [1.04e-04, 1.11e-04, 9.61e-05, 8.64e-06, 9.22e-06, 4.37e-04]    []  
1800      [1.01e-04, 9.77e-05, 9.77e-05, 4.64e-06, 5.55e-06, 3.40e-04]    [9.88e-05, 9.77e-05, 9.77e-05, 4.64e-06, 5.55e-06, 3.40e-04]    []  
1900      [7.88e-05, 8.28e-05, 7.71e-05, 5.28e-06, 5.58e-06, 3.09e-04]    [7.92e-05, 8.28e-05, 7.71e-05, 5.28e-06, 5.58e-06, 3.09e-04]    []  
2000      [2.67e-04, 5.11e-05, 1.17e-05, 2.33e-05, 2.35e-05, 3.73e-04]    [2.96e-04, 5.11e-05, 1.17e-05, 2.33e-05, 2.35e-05, 3.73e-04]    []  
2100      [4.95e-05, 5.93e-05, 5.61e-05, 3.57e-06, 3.66e-06, 2.19e-04]    [4.78e-05, 5.93e-05, 5.61e-05, 3.57e-06, 3.66e-06, 2.19e-04]    []  
2200      [4.66e-05, 4.72e-05, 3.34e-05, 7.45e-06, 3.12e-06, 2.13e-04]    [4.85e-05, 4.72e-05, 3.34e-05, 7.45e-06, 3.12e-06, 2.13e-04]    []  
2300      [3.28e-05, 4.00e-05, 3.78e-05, 2.41e-06, 2.42e-06, 1.47e-04]    [3.08e-05, 4.00e-05, 3.78e-05, 2.41e-06, 2.42e-06, 1.47e-04]    []  
2400      [3.27e-05, 2.77e-05, 2.27e-05, 2.97e-06, 2.99e-06, 1.45e-04]    [3.07e-05, 2.77e-05, 2.27e-05, 2.97e-06, 2.99e-06, 1.45e-04]    []  
2500      [2.58e-05, 2.61e-05, 2.44e-05, 1.60e-06, 1.61e-06, 9.54e-05]    [2.39e-05, 2.61e-05, 2.44e-05, 1.60e-06, 1.61e-06, 9.54e-05]    []  
2600      [2.57e-05, 2.07e-05, 2.20e-05, 1.04e-06, 1.59e-06, 8.15e-05]    [2.10e-05, 2.07e-05, 2.20e-05, 1.04e-06, 1.59e-06, 8.15e-05]    []  
2700      [2.39e-05, 1.68e-05, 1.57e-05, 1.02e-06, 1.05e-06, 6.06e-05]    [2.22e-05, 1.68e-05, 1.57e-05, 1.02e-06, 1.05e-06, 6.06e-05]    []  
2800      [7.51e-05, 1.26e-05, 2.97e-05, 8.28e-08, 3.63e-07, 4.13e-05]    [6.48e-05, 1.26e-05, 2.97e-05, 8.28e-08, 3.63e-07, 4.13e-05]    []  
2900      [2.43e-05, 1.09e-05, 1.01e-05, 6.80e-07, 7.04e-07, 3.89e-05]    [2.28e-05, 1.09e-05, 1.01e-05, 6.80e-07, 7.04e-07, 3.89e-05]    []  
3000      [2.92e-04, 1.05e-06, 7.54e-06, 8.62e-06, 2.31e-05, 9.79e-05]    [2.96e-04, 1.05e-06, 7.54e-06, 8.62e-06, 2.31e-05, 9.79e-05]    []  
3100      [2.49e-05, 7.37e-06, 6.77e-06, 5.02e-07, 5.08e-07, 2.67e-05]    [2.37e-05, 7.37e-06, 6.77e-06, 5.02e-07, 5.08e-07, 2.67e-05]    []  
3200      [2.61e-05, 5.76e-06, 5.20e-06, 3.72e-07, 3.96e-07, 1.99e-05]    [2.49e-05, 5.76e-06, 5.20e-06, 3.72e-07, 3.96e-07, 1.99e-05]    []  
3300      [2.60e-05, 7.25e-06, 6.66e-06, 2.13e-07, 2.88e-07, 1.76e-05]    [2.39e-05, 7.25e-06, 6.66e-06, 2.13e-07, 2.88e-07, 1.76e-05]    []  
3400      [2.56e-05, 4.53e-06, 4.06e-06, 2.94e-07, 3.15e-07, 1.54e-05]    [2.45e-05, 4.53e-06, 4.06e-06, 2.94e-07, 3.15e-07, 1.54e-05]    []  
3500      [2.64e-05, 3.53e-06, 3.09e-06, 2.39e-07, 2.58e-07, 1.19e-05]    [2.53e-05, 3.53e-06, 3.09e-06, 2.39e-07, 2.58e-07, 1.19e-05]    []  
3600      [2.52e-05, 3.44e-06, 3.63e-06, 2.24e-07, 1.84e-07, 1.18e-05]    [2.40e-05, 3.44e-06, 3.63e-06, 2.24e-07, 1.84e-07, 1.18e-05]    []  
3700      [2.54e-05, 2.86e-06, 2.50e-06, 1.90e-07, 2.08e-07, 9.41e-06]    [2.45e-05, 2.86e-06, 2.50e-06, 1.90e-07, 2.08e-07, 9.41e-06]    []  
3800      [1.94e-03, 2.54e-05, 2.44e-04, 4.79e-05, 1.83e-05, 7.79e-05]    [1.85e-03, 2.54e-05, 2.44e-04, 4.79e-05, 1.83e-05, 7.79e-05]    []  
3900      [2.35e-05, 2.85e-06, 2.54e-06, 1.50e-07, 1.92e-07, 8.60e-06]    [2.26e-05, 2.85e-06, 2.54e-06, 1.50e-07, 1.92e-07, 8.60e-06]    []  
4000      [2.38e-05, 2.19e-06, 1.89e-06, 1.44e-07, 1.60e-07, 7.01e-06]    [2.30e-05, 2.19e-06, 1.89e-06, 1.44e-07, 1.60e-07, 7.01e-06]    []  
4100      [4.09e-05, 8.16e-07, 3.44e-08, 1.24e-06, 6.61e-07, 1.13e-05]    [4.10e-05, 8.16e-07, 3.44e-08, 1.24e-06, 6.61e-07, 1.13e-05]    []  
4200      [2.24e-05, 1.77e-06, 1.39e-06, 1.84e-07, 1.69e-07, 7.04e-06]    [2.16e-05, 1.77e-06, 1.39e-06, 1.84e-07, 1.69e-07, 7.04e-06]    []  
4300      [2.24e-05, 1.65e-06, 1.39e-06, 1.08e-07, 1.21e-07, 5.13e-06]    [2.16e-05, 1.65e-06, 1.39e-06, 1.08e-07, 1.21e-07, 5.13e-06]    []  
4400      [9.71e-05, 1.74e-06, 4.34e-06, 3.48e-06, 1.54e-06, 3.54e-05]    [1.01e-04, 1.74e-06, 4.34e-06, 3.48e-06, 1.54e-06, 3.54e-05]    []  
4500      [2.13e-05, 1.54e-06, 1.53e-06, 5.61e-08, 8.09e-08, 4.08e-06]    [2.06e-05, 1.54e-06, 1.53e-06, 5.61e-08, 8.09e-08, 4.08e-06]    []  
4600      [2.02e-05, 1.81e-06, 1.83e-06, 4.57e-08, 9.69e-08, 4.43e-06]    [1.95e-05, 1.81e-06, 1.83e-06, 4.57e-08, 9.69e-08, 4.43e-06]    []  
4700      [1.99e-05, 1.32e-06, 1.09e-06, 8.59e-08, 9.57e-08, 3.97e-06]    [1.93e-05, 1.32e-06, 1.09e-06, 8.59e-08, 9.57e-08, 3.97e-06]    []  
4800      [9.17e-05, 5.59e-06, 1.80e-05, 1.16e-06, 4.53e-07, 2.59e-06]    [8.81e-05, 5.59e-06, 1.80e-05, 1.16e-06, 4.53e-07, 2.59e-06]    []  
4900      [1.83e-05, 1.35e-06, 1.14e-06, 8.80e-08, 9.37e-08, 4.18e-06]    [1.77e-05, 1.35e-06, 1.14e-06, 8.80e-08, 9.37e-08, 4.18e-06]    []  
5000      [1.84e-05, 1.12e-06, 9.02e-07, 7.16e-08, 7.93e-08, 3.27e-06]    [1.78e-05, 1.12e-06, 9.02e-07, 7.16e-08, 7.93e-08, 3.27e-06]    []  

Best model at step 5000:
  train loss: 2.38e-05
  test loss: 2.33e-05
  test metric: []

'train' took 89.635401 s

Compiling model...
'compile' took 2.564832 s

Training model...

Step      Train loss                                                      Test loss                                                       Test metric
5000      [1.84e-05, 1.12e-06, 9.02e-07, 7.16e-08, 7.93e-08, 3.27e-06]    [1.78e-05, 1.12e-06, 9.02e-07, 7.16e-08, 7.93e-08, 3.27e-06]    []  
5100      [7.87e-06, 2.51e-10, 6.58e-09, 6.99e-10, 3.28e-11, 1.25e-07]                                                                        
5176      [2.63e-07, 1.88e-08, 1.75e-09, 8.67e-15, 1.40e-11, 7.65e-08]    [2.67e-07, 1.88e-08, 1.75e-09, 8.67e-15, 1.40e-11, 7.65e-08]    []  

Best model at step 5176:
  train loss: 3.60e-07
  test loss: 3.64e-07
  test metric: []

'train' took 7.904685 s

[lululxvi]

The code looks OK. I am not sure what the issue is.

[forxltk]

q is a function as y but without any BCs. Then the problem may not be well defined. I think you still need left and right BCs of q.

[forxltk]

No, I have tried Poisson equation with only 2-order derivatives. But the result is similar.

If you use external_trainable_variable method to find a scalar q, you have actually told the distribution of q (is a constant) to the NN. Then you will have the only one q. However, without any additional inputs like BC, there could be multiple distribution of the function q for a specific solution.

[tsarikahin]

What do you mean by "Poisson equation with only 2-order derivatives. But the result is similar."? The PDE is already second-order. You mean this example, right?

In the Poisson example, there is no additional information regarding q. But it finds the unknown field correctly.

[forxltk]

Yes. But in this example, q(x) has been a function. While in your code, you use NN to approximate the scalar $q=1$. Then I modified the Poisson example to predict a constant function $q=\pi$.
Here is the pde:

def pde(x, y):
    u, q = y[:, 0:1], y[:, 1:2]
    du_xx = dde.grad.hessian(y, x, component=0, i=0, j=0)
    return -du_xx + q ** 2 * tf.sin(q * x)

And the results:

You can see q is not correct.
The NN knows external_trainable_variable is to predict the scalar, while PFNN is to predict a function. I think it's due to their different implementation. Maybe here is your problem?

[tsarikahin]

Oh no, you are kind of mixing the concept. Just think about q(x)=1. It is a function, right? I can be any function. For simplicity, let's make this clear, it is a function of ones, q(x)=1. I think it is related to higher derivatives, I will show you some results very soon.

[tsarikahin]

I got it now. The NN arch was too complex. I went to Poisson's equation and set u(x)=x**2 so the force is computed as q(x)=1. I run the network and I got weird results. But by changing the number of neurons from 30 to 2, I got the following results. You see force is almost 1, of course, it can be improved. The problem is that if the desired solution is too complex, then approximated solution might be more complex so it fulfills the solution, pde and BCS but it ends up with derivatives to be not correct.

[tsarikahin]

Here is also for Beam example.

I think decreasing the complexity of the model is the key.

[forxltk]

Very good idea! But I still think the problem is not well defined. A larger network would find some other non-linear q that also satisfy the pde and BCs.

[tsarikahin]

You are absolutely right. I figured out one trick. Since the solution is known, one can calculate the 4. derivative of the solution and add them using PointSetBC.

observe_u_xxxx = dde.icbc.PointSetBC(ob_x_temp[:10], ob_u_xxxx, component=1)