PINN with Data
Fluid Mechanics Example
By Prof. Seungchul Lee
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH
http://iai.postech.ac.kr/
Industrial AI Lab at POSTECH
Table of Contents
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import deepxde as dde
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
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from google.colab import drive
drive.mount('/content/drive/')
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fluid_bigdata = np.load('/content/drive/MyDrive/Colab Notebooks/data_files/fluid_bigdata.npy')
observe_x = fluid_bigdata[:, :2]
observe_y = fluid_bigdata[:, 2:]
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observe_u = dde.icbc.PointSetBC(observe_x, observe_y[:, 0].reshape(-1, 1), component=0)
observe_v = dde.icbc.PointSetBC(observe_x, observe_y[:, 1].reshape(-1, 1), component=1)
observe_p = dde.icbc.PointSetBC(observe_x, observe_y[:, 2].reshape(-1, 1), component=2)
1.2. Define Parameters¶
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# Properties
rho = 1
mu = 1
u_in = 1
D = 1
L = 2
1.3. Define Geometry¶
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geom = dde.geometry.Rectangle(xmin = [-L/2, -D/2], xmax = [L/2, D/2])
data = dde.data.PDE(geom,
None,
[observe_u, observe_v, observe_p],
num_domain = 0,
num_boundary = 0,
num_test = 100)
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plt.figure(figsize = (20,4))
plt.scatter(data.train_x_all[:,0], data.train_x_all[:,1], s = 0.5)
plt.scatter(observe_x[:, 0], observe_x[:, 1], c = observe_y[:, 0], s = 6.5, cmap = 'jet')
plt.scatter(observe_x[:, 0], observe_x[:, 1], s = 0.5, color='k', alpha = 0.5)
plt.xlim((0-L/2, L-L/2))
plt.ylim((0-D/2, D-D/2))
plt.xlabel('x-direction length (m)')
plt.ylabel('Distance from middle of plates (m)')
plt.title('Velocity (u)')
plt.show()
1.4. Define Network and Hyper-parameters¶
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layer_size = [2] + [64] * 5 + [3]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
model.compile("adam", lr = 1e-3)
1.5. Train (Adam Optimizer)¶
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losshistory, train_state = model.train(epochs = 10000)
dde.saveplot(losshistory, train_state, issave = False, isplot = False)
1.6. Train More (L-BFGS Optimizer)¶
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dde.optimizers.config.set_LBFGS_options(maxiter=3000)
model.compile("L-BFGS")
losshistory, train_state = model.train()
dde.saveplot(losshistory, train_state, issave = False, isplot = True)
1.7. Plot Results (Adam + L-BFGS)¶
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samples = geom.random_points(500000)
result = model.predict(samples)
color_legend = [[0, 1.5], [-0.3, 0.3], [0, 35]]
for idx in range(3):
plt.figure(figsize = (20, 4))
plt.scatter(samples[:, 0],
samples[:, 1],
c = result[:, idx],
s = 2,
cmap = 'jet')
plt.colorbar()
plt.clim(color_legend[idx])
plt.xlim((0-L/2, L-L/2))
plt.ylim((0-D/2, D-D/2))
plt.tight_layout()
plt.show()
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fluid_smalldata = np.load('/content/drive/MyDrive/Colab Notebooks/data_files/fluid_smalldata.npy')
observe_x = fluid_smalldata[:, :2]
observe_y = fluid_smalldata[:, 2:]
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observe_u = dde.icbc.PointSetBC(observe_x, observe_y[:, 0].reshape(-1, 1), component=0)
observe_v = dde.icbc.PointSetBC(observe_x, observe_y[:, 1].reshape(-1, 1), component=1)
observe_p = dde.icbc.PointSetBC(observe_x, observe_y[:, 2].reshape(-1, 1), component=2)
2.2. Define Geometry¶
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geom = dde.geometry.Rectangle(xmin = [-L/2, -D/2], xmax = [L/2, D/2])
data = dde.data.PDE(geom,
None,
[observe_u, observe_v, observe_p],
num_domain = 0,
num_boundary = 0,
num_test = 120)
In [ ]:
plt.figure(figsize = (20,4))
plt.scatter(data.train_x_all[:,0], data.train_x_all[:,1], s = 0.5)
plt.scatter(observe_x[:, 0], observe_x[:, 1], c = observe_y[:, 0], s = 6.5, cmap = 'jet')
plt.scatter(observe_x[:, 0], observe_x[:, 1], s = 0.5, color='k', alpha = 0.5)
plt.xlim((0-L/2, L-L/2))
plt.ylim((0-D/2, D-D/2))
plt.xlabel('x-direction length (m)')
plt.ylabel('Distance from middle of plates (m)')
plt.title('Velocity (u)')
plt.show()
2.3. Define Network and Hyper-parameters¶
In [ ]:
layer_size = [2] + [64] * 5 + [3]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
model.compile("adam", lr = 1e-3)
2.4. Train (Adam Optimizer)¶
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losshistory, train_state = model.train(epochs = 10000)
dde.saveplot(losshistory, train_state, issave = False, isplot = False)
2.5. Train More (L-BFGS Optimizer)¶
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dde.optimizers.config.set_LBFGS_options(maxiter=3000)
model.compile("L-BFGS")
losshistory, train_state = model.train()
dde.saveplot(losshistory, train_state, issave = False, isplot = True)
2.6. Plot Results (Adam + L-BFGS)¶
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samples = geom.random_points(500000)
result = model.predict(samples)
color_legend = [[0, 1.5], [-0.3, 0.3], [0, 35]]
for idx in range(3):
plt.figure(figsize = (20, 4))
plt.scatter(samples[:, 0],
samples[:, 1],
c = result[:, idx],
s = 2,
cmap = 'jet')
plt.colorbar()
plt.clim(color_legend[idx])
plt.xlim((0-L/2, L-L/2))
plt.ylim((0-D/2, D-D/2))
plt.tight_layout()
plt.show()
3. PINN with Small Data¶
3.1. Define PDE with Boundary & Initial Conditions¶
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def boundary_wall(X, on_boundary):
on_wall = np.logical_and(np.logical_or(np.isclose(X[1], -D/2), np.isclose(X[1], D/2)), on_boundary)
return on_wall
def boundary_inlet(X, on_boundary):
return on_boundary and np.isclose(X[0], -L/2)
def boundary_outlet(X, on_boundary):
return on_boundary and np.isclose(X[0], L/2)
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def pde(X, Y):
du_x = dde.grad.jacobian(Y, X, i = 0, j = 0)
du_y = dde.grad.jacobian(Y, X, i = 0, j = 1)
dv_x = dde.grad.jacobian(Y, X, i = 1, j = 0)
dv_y = dde.grad.jacobian(Y, X, i = 1, j = 1)
dp_x = dde.grad.jacobian(Y, X, i = 2, j = 0)
dp_y = dde.grad.jacobian(Y, X, i = 2, j = 1)
du_xx = dde.grad.hessian(Y, X, i = 0, j = 0, component = 0)
du_yy = dde.grad.hessian(Y, X, i = 1, j = 1, component = 0)
dv_xx = dde.grad.hessian(Y, X, i = 0, j = 0, component = 1)
dv_yy = dde.grad.hessian(Y, X, i = 1, j = 1, component = 1)
pde_u = Y[:,0:1] * du_x + Y[:,1:2] * du_y + 1/rho * dp_x - (mu/rho) * (du_xx + du_yy)
pde_v = Y[:,0:1] * dv_x + Y[:,1:2] * dv_y + 1/rho * dp_y - (mu/rho) * (dv_xx + dv_yy)
pde_cont = du_x + dv_y
return [pde_u, pde_v, pde_cont]
3.2. Define Geometry and Implement Boundary Condition¶
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geom = dde.geometry.Rectangle(xmin=[-L/2, -D/2], xmax=[L/2, D/2])
bc_wall_u = dde.DirichletBC(geom, lambda X: 0., boundary_wall, component = 0)
bc_wall_v = dde.DirichletBC(geom, lambda X: 0., boundary_wall, component = 1)
bc_inlet_u = dde.DirichletBC(geom, lambda X: u_in, boundary_inlet, component = 0)
bc_inlet_v = dde.DirichletBC(geom, lambda X: 0., boundary_inlet, component = 1)
bc_outlet_p = dde.DirichletBC(geom, lambda X: 0., boundary_outlet, component = 2)
bc_outlet_v = dde.DirichletBC(geom, lambda X: 0., boundary_outlet, component = 1)
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data = dde.data.PDE(geom,
pde,
[bc_wall_u, bc_wall_v, bc_inlet_u, bc_inlet_v, bc_outlet_p, bc_outlet_v, observe_u, observe_v, observe_p],
num_domain = 1000,
num_boundary = 500,
num_test = 1000,
train_distribution = 'LHS')
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plt.figure(figsize = (20,4))
plt.scatter(data.train_x_all[:,0], data.train_x_all[:,1], s = 0.5)
plt.scatter(observe_x[:, 0], observe_x[:, 1], c = observe_y[:, 0], s = 6.5, cmap = 'jet')
plt.scatter(observe_x[:, 0], observe_x[:, 1], s = 0.5, color='k', alpha = 0.5)
plt.xlim((0-L/2, L-L/2))
plt.ylim((0-D/2, D-D/2))
plt.xlabel('x-direction length (m)')
plt.ylabel('Distance from middle of plates (m)')
plt.title('Velocity (u)')
plt.show()
3.3. Define Network and Hyper-parameters¶
In [ ]:
layer_size = [2] + [64] * 5 + [3]
activation = "tanh"
initializer = "Glorot uniform"
net = dde.maps.FNN(layer_size, activation, initializer)
model = dde.Model(data, net)
model.compile("adam", lr = 1e-3, loss_weights = [1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 9])
3.4. Train (Adam Optimizer)¶
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losshistory, train_state = model.train(epochs = 10000)
dde.saveplot(losshistory, train_state, issave = False, isplot = False)
3.5. Train More (L-BFGS Optimizer)¶
In [ ]:
dde.optimizers.config.set_LBFGS_options(maxiter=3000)
model.compile("L-BFGS", loss_weights = [1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 9])
losshistory, train_state = model.train()
dde.saveplot(losshistory, train_state, issave = False, isplot = True)
3.6. Plot Results (Adam + L-BFGS)¶
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samples = geom.random_points(500000)
result = model.predict(samples)
color_legend = [[0, 1.5], [-0.3, 0.3], [0, 35]]
for idx in range(3):
plt.figure(figsize = (20, 4))
plt.scatter(samples[:, 0],
samples[:, 1],
c = result[:, idx],
s = 2,
cmap = 'jet')
plt.colorbar()
plt.clim(color_legend[idx])
plt.xlim((0-L/2, L-L/2))
plt.ylim((0-D/2, D-D/2))
plt.tight_layout()
plt.show()
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%%javascript
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