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
from torch.utils.cpp_extension import load_inline

… CUDA C++ source code for the kernel …
relu_source = “”"
…
“”"

relu_cpp_source = “”"
torch::Tensor relu_cuda(torch::Tensor x);
“”"

Compile the inline CUDA code
relu = load_inline(
name=“relu”,
cpp_sources=relu_cpp_source,
cuda_sources=relu_source,
functions=[“relu_cuda”],
verbose=True
)

class ModelNew(torch.nn.Module):
def init(self):
super(ModelNew, self).init()
self.relu = relu # The module containing the kernel

def forward(self, x):
    return self.relu.relu_cuda(x)


You are given the following architecture:   
  
python
import torch
import torch.nn as nn

class Model(nn.Module):
“”"
Simple model that performs a LogSumExp activation over the last dimension.
“”"
def init(self):
super(Model, self).init()

def forward(self, x: torch.Tensor) -> torch.Tensor:  
    """  
    Applies LogSumExp activation to the input tensor.

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

    Returns:  
        torch.Tensor: Output tensor of shape [batch_size].  
    """  
    return torch.logsumexp(x, dim=-1)  
batch_size = 16
dim = 16384

def get_inputs():
x = torch.randn(batch_size, dim)
return [x]

def get_init_inputs():
return [] # No special initialization inputs needed

Your task is to write a new file `logsumexp_cudacode.py` that defines a new model `ModelNew` which uses a custom CUDA kernel to accelerate the `torch.logsumexp` operation. The goal is to achieve a significant speedup while maintaining numerical precision. The implementation should be robust and handle the input dimensions correctly. The final output should be a single python file containing the CUDA kernel, the compilation logic via `load_inline`, and the new `ModelNew` class.
