Pytorch AutoGrad Functions¶
SISO¶
In [ ]:
import torch
from torch.autograd import grad
x = torch.Tensor([2])
x.requires_grad = True
def model(x): return x ** 3
y = model(x)
dy_dx = grad(outputs= y, inputs=x)
print(f"x: \n {x} \n y=x^3: \n {y} \n dy_dx:\n {dy_dx} , 3*x^2: \n {3*x**2}")
# 3x²
x:
tensor([2.], requires_grad=True)
y=x^3:
tensor([8.], grad_fn=<PowBackward0>)
dy_dx:
(tensor([12.]),)
MISO¶
In [2]:
import torch
from torch.autograd import grad
import traceback
x = torch.Tensor([2, 3, 4])
x.requires_grad = True
def model(x): return x ** 3
print(x)
y = model(x)
print(y)
try:
dy_dx = grad(outputs= y, inputs=x)
print(f"x: \n {x} \n y=x^3: \n {y} \n dy_dx:\n {dy_dx}")
except Exception as e:
print(f"!!!!!!!!!!!!!!! ERROR: {e}")
traceback.print_exc()
tensor([2., 3., 4.], requires_grad=True) tensor([ 8., 27., 64.], grad_fn=<PowBackward0>) !!!!!!!!!!!!!!! ERROR: grad can be implicitly created only for scalar outputs
Traceback (most recent call last):
File "/tmp/ipykernel_1955206/3879111839.py", line 15, in <module>
dy_dx = grad(outputs= y, inputs=x)
File "/home/bneveu/.local/lib/python3.10/site-packages/torch/autograd/__init__.py", line 367, in grad
grad_outputs_ = _make_grads(
File "/home/bneveu/.local/lib/python3.10/site-packages/torch/autograd/__init__.py", line 117, in _make_grads
raise RuntimeError(
RuntimeError: grad can be implicitly created only for scalar outputs
MISO - Loop¶
In [3]:
import torch
from torch.autograd import grad
x = torch.Tensor([2, 3, 4])
x.requires_grad = True
def model(x): return x ** 3
y = model(x)
test = torch.autograd.grad(outputs=y[0], inputs=x, retain_graph=True)
print('test: ' + str(test))
for i, out in enumerate(y):
dy_dx = torch.autograd.grad(outputs=out, inputs=x, retain_graph=True)
print(dy_dx)
dy_dx = [torch.autograd.grad(outputs=out, inputs=x, retain_graph=True)[0][i]
for i, out in enumerate(y)]
#print(f"x: \n {x} \n y=x^3: \n {y} \n dy_dx:\n {dy_dx}")
test: (tensor([12., 0., 0.]),) (tensor([12., 0., 0.]),) (tensor([ 0., 27., 0.]),) (tensor([ 0., 0., 48.]),)
In [4]:
import torch
from torch.autograd import grad
x = torch.Tensor([2, 3, 4])
x.requires_grad = True
def model(x): return x ** 3
y = model(x)
test = torch.autograd.grad(outputs=y[0], inputs=x, retain_graph=True)
print('test: ' + str(test))
for i, out in enumerate(y):
test = torch.autograd.grad(outputs=out, inputs=x, retain_graph=True)
print(test)
# dy_dx = [torch.autograd.grad(outputs=out, inputs=x, retain_graph=True)[0][i]
# for i, out in enumerate(y)]
print(f"x: \n {x} \n y=x^3: \n {y} \n dy_dx:\n {dy_dx}")
test: (tensor([12., 0., 0.]),)
(tensor([12., 0., 0.]),)
(tensor([ 0., 27., 0.]),)
(tensor([ 0., 0., 48.]),)
x:
tensor([2., 3., 4.], requires_grad=True)
y=x^3:
tensor([ 8., 27., 64.], grad_fn=<PowBackward0>)
dy_dx:
[tensor(12.), tensor(27.), tensor(48.)]
MISO - Loop with Large Data¶
:warning: Pretty SLOW ->
Use the sum trick
In [1]:
import torch
from torch.autograd import grad
x = torch.Tensor(range(400))
x.requires_grad = True
def model(x): return x ** 3
y = model(x)
dy_dx = [torch.autograd.grad(outputs=out, inputs=x, retain_graph=True)[0][i]
for i, out in enumerate(y)]
#3 seconds for 400 elements
#print(f"x: \n {x} \n y=x^3: \n {y} \n dy_dx:\n {dy_dx}")
MISO - Sum¶
In [6]:
import torch
from torch.autograd import grad
x = torch.Tensor([2, 3, 4])
x.requires_grad = True
def model(x): return x ** 3
y = model(x)
dy_dx = torch.autograd.grad(outputs=y.sum(), inputs=x, retain_graph=True)[0]
print(f"x: \n {x} \n y=x^3: \n {y} \n dy_dx:\n {dy_dx}")
x:
tensor([2., 3., 4.], requires_grad=True)
y=x^3:
tensor([ 8., 27., 64.], grad_fn=<PowBackward0>)
dy_dx:
tensor([12., 27., 48.])
In [8]:
import torch
from torch.autograd import grad
x = torch.Tensor(range(40000))
x.requires_grad = True
def model(x): return x ** 3
y = model(x)
dy_dx = torch.autograd.grad(outputs=y.sum(), inputs=x, retain_graph=True)[0]
print(dy_dx[:20])
tensor([ 0., 3., 12., 27., 48., 75., 108., 147., 192., 243.,
300., 363., 432., 507., 588., 675., 768., 867., 972., 1083.])
MIMO - Jacobian¶
In [9]:
from torch.autograd.functional import jacobian
from torch import tensor
#Defining the main function
def f(x1,x2,x3):
x23 = x2**3
x31 = x3*x1
return (x1.sum() + x23.sum(), x31.sum())
x1 = torch.Tensor(range(400))
x2 = 2*torch.Tensor(range(400))
x3 = 3*torch.Tensor(range(400))
x = (x1, x2, x3)
#Printing the Jacobian
jac = jacobian(f,x)
print(len(jac))
print(jac[0][0].size())
2 torch.Size([400])
In [15]:
from torch.autograd.functional import jacobian
from torch import tensor
#Defining the main function
def f(x1,x2,x3):
return (x1 + x2**3, x3*x1, x2**2*x3+x1)
#Defining input tensors
x1 = tensor(3.0)
x2 = tensor(4.0)
x3 = tensor(5.0)
#Printing the Jacobian
print(jacobian(f,(x1,x2,x3)))
# 4 outputs , 3 inputs
# -> 4x3 matrix
((tensor(1.), tensor(48.), tensor(0.)), (tensor(5.), tensor(0.), tensor(3.)), (tensor(1.), tensor(40.), tensor(16.)))
MIMO - Hessian¶
In [16]:
from torch.autograd.functional import hessian
from torch import tensor
#Defining the main function
def f(x1,x2,x3):
return x1**3+2*x2**3+x1*x3**2
#Defining input tensors
x1 = tensor(3.0)
x2 = tensor(4.0)
x3 = tensor(5.0)
H = hessian(f,(x1,x2,x3))
print(H)
((tensor(18.), tensor(0.), tensor(10.)), (tensor(0.), tensor(48.), tensor(0.)), (tensor(10.), tensor(0.), tensor(6.)))
Example problem:¶
source: https://github.com/udemirezen/PINN-1/blob/main/solve_PDE_NN.ipynb
We have been given a PDE: du/dx=2du/dt+u and boundary condition: u(x,0)=6e^(-3x)
Independent variables: x,t (input) Dependent variables: u (outputs) We have to find out u(x,t) for all x in range [0,2] and t in range [0,1]
When we solved this problem analytically, we found the solution: u(x,t) = 6e^(-3x-2t)
Our f is f = du/dx - 2du/dt - u
In [17]:
import torch
import torch.nn as nn
from torch.autograd import Variable
device = torch.device("cuda:0" if torch.cuda.is_available() else "cpu")
import numpy as np
In [18]:
# define NN:
class Net(nn.Module):
def __init__(self):
super(Net, self).__init__()
self.hidden_layer1 = nn.Linear(2,5)
self.hidden_layer2 = nn.Linear(5,5)
self.hidden_layer3 = nn.Linear(5,5)
self.hidden_layer4 = nn.Linear(5,5)
self.hidden_layer5 = nn.Linear(5,5)
self.output_layer = nn.Linear(5,1)
def forward(self, x,t):
inputs = torch.cat([x,t],axis=1) # combined two arrays of 1 columns each to one array of 2 columns
layer1_out = torch.sigmoid(self.hidden_layer1(inputs))
layer2_out = torch.sigmoid(self.hidden_layer2(layer1_out))
layer3_out = torch.sigmoid(self.hidden_layer3(layer2_out))
layer4_out = torch.sigmoid(self.hidden_layer4(layer3_out))
layer5_out = torch.sigmoid(self.hidden_layer5(layer4_out))
output = self.output_layer(layer5_out) ## For regression, no activation is used in output layer
return output
In [19]:
### (2) Model
net = Net()
net = net.to(device)
mse_cost_function = torch.nn.MSELoss() # Mean squared error
optimizer = torch.optim.Adam(net.parameters())
In [20]:
## PDE as loss function
def f(x,t, net):
u = net(x,t)
## Based on our f = du/dx - 2du/dt - u, we need du/dx and du/dt
u_x = torch.autograd.grad(u.sum(), x, create_graph=True)[0]
u_t = torch.autograd.grad(u.sum(), t, create_graph=True)[0]
pde = u_x - 2*u_t - u
return pde
In [21]:
## Data from Boundary Conditions for NN
x_bc = np.random.uniform(low=0.0, high=2.0, size=(500,1))
t_bc = np.zeros((500,1))
u_bc = 6*np.exp(-3*x_bc)
## Data generated for entire PINN
x_all = np.random.uniform(low=0.0, high=2.0, size=(500,1))
t_all = np.random.uniform(low=0.0, high=1.0, size=(500,1))
all_zeros = np.zeros((500,1))
In [22]:
### (3) Training / Fitting
iterations = 5000
for epoch in range(iterations):
optimizer.zero_grad()
# Loss based on boundary conditions
pt_x_bc = Variable(torch.from_numpy(x_bc).float(), requires_grad=False).to(device)
pt_t_bc = Variable(torch.from_numpy(t_bc).float(), requires_grad=False).to(device)
pt_u_bc = Variable(torch.from_numpy(u_bc).float(), requires_grad=False).to(device)
net_bc_out = net(pt_x_bc, pt_t_bc) # output of u(x,t)
mse_u = mse_cost_function(net_bc_out, pt_u_bc)
# Loss based on PDE
pt_x_all = Variable(torch.from_numpy(x_all).float(), requires_grad=True).to(device)
pt_t_all = Variable(torch.from_numpy(t_all).float(), requires_grad=True).to(device)
pt_all_zeros = Variable(torch.from_numpy(all_zeros).float(), requires_grad=False).to(device)
f_out = f(pt_x_all, pt_t_all, net) # output of f(x,t)
mse_f = mse_cost_function(f_out, pt_all_zeros)
# Combining the loss functions
loss = mse_u + mse_f
loss.backward()
optimizer.step()
with torch.autograd.no_grad():
print(epoch,"Traning Loss:",loss.data)
0 Traning Loss: tensor(2.3730, device='cuda:0') 1 Traning Loss: tensor(2.3680, device='cuda:0') 2 Traning Loss: tensor(2.3632, device='cuda:0') 3 Traning Loss: tensor(2.3584, device='cuda:0') 4 Traning Loss: tensor(2.3537, device='cuda:0') 5 Traning Loss: tensor(2.3492, device='cuda:0') 6 Traning Loss: tensor(2.3447, device='cuda:0') 7 Traning Loss: tensor(2.3404, device='cuda:0') 8 Traning Loss: tensor(2.3362, device='cuda:0') 9 Traning Loss: tensor(2.3320, device='cuda:0') 10 Traning Loss: tensor(2.3281, device='cuda:0') 11 Traning Loss: tensor(2.3242, device='cuda:0') 12 Traning Loss: tensor(2.3204, device='cuda:0') 13 Traning Loss: tensor(2.3168, device='cuda:0') 14 Traning Loss: tensor(2.3132, device='cuda:0') 15 Traning Loss: tensor(2.3098, device='cuda:0') 16 Traning Loss: tensor(2.3065, device='cuda:0') 17 Traning Loss: tensor(2.3034, device='cuda:0') 18 Traning Loss: tensor(2.3003, device='cuda:0') 19 Traning Loss: tensor(2.2973, device='cuda:0') 20 Traning Loss: 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tensor(0.0007, device='cuda:0') 4980 Traning Loss: tensor(0.0007, device='cuda:0') 4981 Traning Loss: tensor(0.0007, device='cuda:0') 4982 Traning Loss: tensor(0.0007, device='cuda:0') 4983 Traning Loss: tensor(0.0007, device='cuda:0') 4984 Traning Loss: tensor(0.0007, device='cuda:0') 4985 Traning Loss: tensor(0.0007, device='cuda:0') 4986 Traning Loss: tensor(0.0007, device='cuda:0') 4987 Traning Loss: tensor(0.0007, device='cuda:0') 4988 Traning Loss: tensor(0.0007, device='cuda:0') 4989 Traning Loss: tensor(0.0007, device='cuda:0') 4990 Traning Loss: tensor(0.0007, device='cuda:0') 4991 Traning Loss: tensor(0.0007, device='cuda:0') 4992 Traning Loss: tensor(0.0007, device='cuda:0') 4993 Traning Loss: tensor(0.0007, device='cuda:0') 4994 Traning Loss: tensor(0.0007, device='cuda:0') 4995 Traning Loss: tensor(0.0007, device='cuda:0') 4996 Traning Loss: tensor(0.0007, device='cuda:0') 4997 Traning Loss: tensor(0.0007, device='cuda:0') 4998 Traning Loss: tensor(0.0007, device='cuda:0') 4999 Traning Loss: tensor(0.0007, device='cuda:0')
In [ ]:
Plot Results¶
In [23]:
import pandas as pd
import plotly.express as px
pd.options.plotting.backend = "plotly"
In [24]:
x=np.arange(0,2,0.02)
t=np.arange(0,1,0.02)
ms_x, ms_t = np.meshgrid(x, t)
## Just because meshgrid is used, we need to do the following adjustment
x = np.ravel(ms_x).reshape(-1,1)
t = np.ravel(ms_t).reshape(-1,1)
pt_x = Variable(torch.from_numpy(x).float(), requires_grad=True).to(device)
pt_t = Variable(torch.from_numpy(t).float(), requires_grad=True).to(device)
pt_u = net(pt_x,pt_t)
u=pt_u.data.cpu().numpy()
In [25]:
dict_data = {"x":ms_x.flatten(),"t":ms_t.flatten(),"u":u.flatten(), "u_analytical": 6*np.exp(-3*ms_x.flatten()-2*ms_t.flatten())}
df = pd.DataFrame(dict_data)
In [26]:
fig = px.scatter_3d(df, x='x', y='t', z='u',color="u")
fig.show()
In [27]:
fig = px.scatter_3d(df, x='x', y='t', z='u_analytical',color="u_analytical")
fig.show()