You write custom CUDA kernels to replace the pytorch operators in the given architecture to get speedups.  

You have complete freedom to choose the set of operators you want to replace. You may make the decision to replace some operators with custom CUDA kernels and leave others unchanged. You may replace multiple operators with custom implementations, consider operator fusion opportunities (combining multiple operators into a single kernel, for example, combining matmul+relu), or algorithmic changes (such as online softmax). You are only limited by your imagination.  

Here's an example to show you the syntax of inline embedding custom CUDA operators in torch: The example given architecture is:  

python
import torch
import torch.nn as nn
import torch.nn.functional as F

class Model(nn.Module):
def init(self) -> None:
super().init()

def forward(self, a, b):
    return a + b
def get_inputs():
# randomly generate input tensors based on the model architecture
a = torch.randn(1, 128).cuda()
b = torch.randn(1, 128).cuda()
return [a, b]

def get_init_inputs():
# randomly generate tensors required for initialization based on the model architecture
return []



The example new arch with custom CUDA kernels looks like this:  

python
import torch
import torch.nn as nn
import torch.nn.functional as F

class Model(nn.Module):
def init(self) -> None:
super().init()

def forward(self, a, b):
    return a + b
def get_inputs():
# randomly generate input tensors based on the model architecture
a = torch.randn(1, 128).cuda()
b = torch.randn(1, 128).cuda()
return [a, b]

def get_init_inputs():
# randomly generate tensors required for initialization based on the model architecture
return []



You are given the following architecture:  

python
import torch
import torch.nn as nn

class Model(nn.Module):
“”"
Simple model that performs a GroupNorm operation.
“”"
def init(self, num_channels=64, num_groups=8, eps=1e-5, affine=True):
super(Model, self).init()
self.num_channels = num_channels
self.num_groups = num_groups
self.eps = eps
self.affine = affine

    # Create GroupNorm layer
    self.group_norm = nn.GroupNorm(
        num_groups=num_groups,
        num_channels=num_channels,
        eps=eps,
        affine=affine
    )
  
def forward(self, x: torch.Tensor) -> torch.Tensor:  
    """  
    Applies GroupNorm to the input tensor.  

    Args:  
        x (torch.Tensor): Input tensor of shape [batch_size, num_channels, height, width]  

    Returns:  
        torch.Tensor: Output tensor after group normalization, same shape as input.  
    """  
    return self.group_norm(x)  
batch_size = 8
num_channels = 64
height = 32
width = 32
num_groups = 8

def get_inputs():
x = torch.randn(batch_size, num_channels, height, width)
return [x]

def get_init_inputs():
return [num_channels, num_groups] # GroupNorm needs num_channels and num_groups



IMPORTANT REQUIREMENTS:
1. Maintain strict float32 precision for all computations
2. Ensure numerical stability in variance calculation
3. Optimize for MACA architecture compatibility
4. Use efficient memory access patterns (vectorized loading when possible)
5. Implement warp-level reductions for better performance
6. Handle edge cases and boundary conditions properly
7. Maintain exact numerical alignment with PyTorch's GroupNorm implementation