Write pytorch-based Python code to implement a neural network that solves a regression problem with an output layer of a positively weighted sub-network plus a negatively weighted sub-network， where the negative weights are random and take the opposite of the absolute value of the standard normal distribution, the exact value of the weights are obtained from the objective function by Back propagation.

Sure! Here's an example of PyTorch-based code that implements a neural network for a regression problem with a positively weighted sub-network and a negatively weighted sub-network:

import torch
import torch.nn as nn
import torch.optim as optim

# Define the neural network architecture
class RegressionNet(nn.Module):
    def __init__(self, input_size, hidden_size):
        super(RegressionNet, self).__init__()
        self.positive_net = nn.Linear(input_size, hidden_size)
        self.negative_net = nn.Linear(input_size, hidden_size)

    def forward(self, x):
        positive_output = self.positive_net(x)
        negative_output = self.negative_net(x)
        output = positive_output - negative_output
        return output

# Set random seed for reproducibility
torch.manual_seed(42)

# Define the hyperparameters
input_size = 10
hidden_size = 20
learning_rate = 0.01
num_epochs = 1000

# Generate random input and output data
x = torch.randn(100, input_size)
y = torch.randn(100, 1)

# Initialize the regression network
model = RegressionNet(input_size, hidden_size)

# Define the loss function and optimizer
criterion = nn.MSELoss()
optimizer = optim.SGD(model.parameters(), lr=learning_rate)

# Training loop
for epoch in range(num_epochs):
    # Forward pass and compute the loss
    outputs = model(x)
    loss = criterion(outputs, y)

    # Backward pass and optimize the model
    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

    # Print the loss every 100 epochs
    if (epoch+1) % 100 == 0:
        print(f'Epoch: {epoch+1}/{num_epochs}, Loss: {loss.item():.4f}')

# Test the trained model
x_test = torch.randn(10, input_size)
with torch.no_grad():
    predicted = model(x_test)
    print(f'Predicted values: {predicted.squeeze().tolist()}')

In this code, we define a `RegressionNet` class that inherits from `nn.Module` and contains two linear layers: `positive_net` and `negative_net`. The forward pass subtracts the output of the negative network from the positive network to obtain the final output.

We then use mean squared error (MSE) loss as the objective function and stochastic gradient descent (SGD) as the optimizer. The model is trained for a specified number of epochs, and the loss is printed every 100 epochs. Finally, we test the trained model on a small test dataset.

Please note that the exact weights are obtained through backpropagation during the training process, and the weights are not directly initialized from the objective function.