How to get a better result?

[whtwu]

Hello, @lululxvi , I have tried many ways to optimize the model ,unfortunately, I cant't get a more accurate result, I would be grateful if there are any suggestions for me ,thanks a lot!

`import deepxde as dde import numpy as np from deepxde.backend import tf import matplotlib.pyplot as plt

def main(): b = 0.06 h = 0.002 e = 2e+11 i = 4 * pow(10, -11) rho = 7850 s = 12e-05 l = 1 f = 10 omiga = 1

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

def ddt(x, y):
    return dde.grad.hessian(y, x, i=0, j=1)

def pde(x, y):
    f_xt = f * tf.sin(omiga * x[:, 1:])
    dy_tt = ddt(x,y)
    dy_xx = ddx(x,y)
    dy_xxxx = dde.grad.hessian(dy_xx, x, i=0, j=0)
    return e * i * dy_xxxx + rho * s * dy_tt - f_xt

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], l)

def func(x):
    sum = 0
    for n in range(1,22):
        omiga_i = ((n ** 2 * np.pi ** 2) / l ** 2) * (e * i / (rho * s)) ** 0.5
        solve = (2 / (rho * s * l) * f / (omiga_i ** 2 - omiga ** 2) * 2 * l / (n * np.pi) * \
        np.sin((n * np.pi * x[:, 0:1]) / l) * (np.sin(omiga * x[:, 1:]) - omiga / omiga_i * np.sin(omiga_i * x[:, 1:])))
        sum += solve
    return sum

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

# bc1 = dde.DirichletBC(geomtime, lambda x: 0, lambda _, on_boundary: on_boundary)
# bc2 = dde.OperatorBC(geomtime, lambda x, y, _: ddx(x, y), lambda _, on_boundary: on_boundary)

bc1 = dde.DirichletBC(geomtime, lambda x: 0, boundary_l)
bc2 = dde.OperatorBC(geomtime, lambda x, y, _: ddx(x, y), boundary_l)
bc3 = dde.DirichletBC(geomtime, lambda x: 0, boundary_r)
bc4 = dde.OperatorBC(geomtime, lambda x, y, _: ddx(x, y), boundary_r)

# x_ = np.linspace(0.3, 0.7, num=200)
# t_ = np.linspace(0, 1,10)
# xx_,tt_ = np.meshgrid(x_,t_)
# observe_x = np.vstack((np.ravel(xx_), np.ravel(tt_))).T

data = dde.data.TimePDE(
    geomtime,
    pde,
    [bc1, bc2, bc3, bc4],
    num_domain=400,
    num_boundary=60,
    #anchors=observe_x,
    #train_distribution='uniform',
    solution=func,
    num_test=4000,
)

layer_size = [2] + [50] * 4 + [1]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.nn.FNN(layer_size, activation, initializer)

#net = dde.nn.STMsFFN(layer_size, activation, initializer, sigmas_x=[1], sigmas_t=[1, 10])

net.apply_output_transform(lambda x, y:  x[:,0:1] * (1 - x[:,0:1]) * x[:,1:] * y)
model = dde.Model(data, net)
model.compile("adam", lr=0.001, metrics=["l2 relative error"])

#pde_residual_resampler = dde.callbacks.PDEResidualResampler(period=1)

model.train(epochs=20000, display_every=2000)
#model.train(epochs=20000, callbacks=[pde_residual_resampler], display_every=2000)

model.compile("L-BFGS",  metrics=["l2 relative error"])
losshistory, train_state = model.train()
dde.saveplot(losshistory, train_state, issave=True, isplot=True)

x = np.linspace(0, 1, 20, endpoint=True)
t = [0.2]
xx, tt = np.meshgrid(x, t)
X = np.vstack((np.ravel(xx), np.ravel(tt))).T
y_pred = model.predict(X)
y_exact = func(X)
plt.plot(x,y_pred, color="blue", linewidth=2, linestyle="solid", label="predict")
plt.plot(x,func(X), color="red", linewidth=2.5, linestyle="dotted", label="exact")
plt.legend()
plt.xlabel("x")
plt.xticks(np.arange(0,1.1,0.1))
plt.ylabel("y")
plt.title("t=0.2s")
plt.show()

if name == "main": main() ` Step Train loss Test loss Test metric
0 [1.14e+01, 0.00e+00, 3.05e-02, 0.00e+00, 1.34e+00] [8.76e+00, 0.00e+00, 3.05e-02, 0.00e+00, 1.34e+00] [7.24e+00]
2000 [2.36e-04, 0.00e+00, 1.88e-04, 0.00e+00, 2.12e-04] [1.68e-04, 0.00e+00, 1.88e-04, 0.00e+00, 2.12e-04] [6.91e-02]
4000 [1.05e-04, 0.00e+00, 5.39e-05, 0.00e+00, 2.02e-05] [8.56e-05, 0.00e+00, 5.39e-05, 0.00e+00, 2.02e-05] [6.07e-02]
6000 [4.67e-04, 0.00e+00, 3.83e-05, 0.00e+00, 2.52e-05] [3.57e-04, 0.00e+00, 3.83e-05, 0.00e+00, 2.52e-05] [6.91e-02]
8000 [5.68e-05, 0.00e+00, 1.58e-05, 0.00e+00, 1.69e-06] [4.85e-05, 0.00e+00, 1.58e-05, 0.00e+00, 1.69e-06] [6.67e-02]
10000 [6.12e-05, 0.00e+00, 1.81e-05, 0.00e+00, 1.54e-06] [4.97e-05, 0.00e+00, 1.81e-05, 0.00e+00, 1.54e-06] [6.67e-02]
12000 [2.41e-04, 0.00e+00, 2.88e-05, 0.00e+00, 1.84e-05] [2.42e-04, 0.00e+00, 2.88e-05, 0.00e+00, 1.84e-05] [9.14e-02]
14000 [4.24e-05, 0.00e+00, 1.62e-05, 0.00e+00, 3.11e-06] [3.57e-05, 0.00e+00, 1.62e-05, 0.00e+00, 3.11e-06] [6.79e-02]
16000 [3.89e-05, 0.00e+00, 1.60e-05, 0.00e+00, 3.17e-06] [3.30e-05, 0.00e+00, 1.60e-05, 0.00e+00, 3.17e-06] [6.78e-02]
18000 [2.43e-04, 0.00e+00, 3.28e-05, 0.00e+00, 9.17e-05] [2.15e-04, 0.00e+00, 3.28e-05, 0.00e+00, 9.17e-05] [1.21e-01]
20000 [3.43e-05, 0.00e+00, 1.44e-05, 0.00e+00, 3.45e-06] [2.92e-05, 0.00e+00, 1.44e-05, 0.00e+00, 3.45e-06] [6.75e-02]

Best model at step 20000: train loss: 5.21e-05 test loss: 4.70e-05 test metric: [6.75e-02]

'train' took 786.180204 s

Compiling model... 'compile' took 4.721393 s

Training model...

Step Train loss Test loss Test metric
20000 [3.43e-05, 0.00e+00, 1.44e-05, 0.00e+00, 3.45e-06] [2.92e-05, 0.00e+00, 1.44e-05, 0.00e+00, 3.45e-06] [6.75e-02]
21000 [5.80e-06, 0.00e+00, 3.88e-06, 0.00e+00, 2.78e-06]
21915 [4.72e-06, 0.00e+00, 2.21e-06, 0.00e+00, 1.93e-06] [3.49e-06, 0.00e+00, 2.21e-06, 0.00e+00, 1.93e-06] [5.94e-02]

Best model at step 21915: train loss: 8.86e-06 test loss: 7.63e-06 test metric: [5.94e-02]

[lululxvi]

Have you checked FAQ "Q: I failed to train the network or get the right solution, e.g., large training loss, unbalanced losses."

[whtwu]

Thanks for your reply. Absoulately yes, I have checked FAQ and other issues. and the time domain is (0,1), the result gets better as the closer t is to 1.

[whtwu]

@lululxvi ，Looking forward to your suggestion.

[lululxvi]

You may try hard constraints for the BC.

[engsbk]

Is there any update on this issue? I want to know if the results improved when using hard constraints for BC.

I'm having the same problem, and I did not resolve to use BC because I found it hard to represent my initial and boundary conditions with hard constraints without negatively impacting my results.

Looking forward to completing this issue discussion!

[whtwu]

This is my BC: In the above code, I used the hard constraint to satisfy the Dirichlet BC,, net.apply_output_transform(lambda x, y: x[:,0:1] * (1 - x[:,0:1]) * x[:,1:] * y) but I don't know how to satisfy the OperatorBC at the same time.