You write custom CUDA kernels to replace the pytorch operators in the given GeGLU 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 chunk+gelu+elementwise_mul), or algorithmic changes (such as optimized memory access patterns). You are only limited by your imagination.
This code implements Wasserstein distance + energy statistic + GELU activation with CUDA optimizations:

In-place bitonic sort - Parallel bitonic sorting algorithm for both x and target vectors in shared memory.

Shared memory for sorting - Uses dynamic shared memory allocation (2×width) for parallel sorting.

Parallel sorting - Threads cooperate to sort both vectors using bitonic merge networks.

Energy distance computation - Computes squared differences between sorted vectors (1D Wasserstein-2).

Warp/block reduction - Standard parallel reduction for sum of squared differences.

Fused activation - Applies GELU to the normalized energy distance.

Cooperative sorting - Multiple synchronization points (__syncthreads()) for sorting phases.

Assumed power-of-2 width - Bitonic sort requires width to be power of 2.

Memory coalescing - Global memory loads are contiguous, then sorting happens in shared memory.

Batch parallelism - One CUDA block per input row with large shared memory allocation.




Here's an example to show you the syntax of inline embedding custom CUDA operators in torch: The example given architecture is:
import torch
import torch.nn as nn
import torch.nn.functional as F


class Model(nn.Module):
    def __init__(self, target):
        super(Model, self).__init__()
        self.target = nn.Parameter(target)

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        x_sorted, _ = torch.sort(x, dim=-1)
        target_sorted, _ = torch.sort(self.target, dim=-1)

        diff = x_sorted - target_sorted
        energy = torch.mean(diff * diff, dim=-1)

        return F.gelu(energy)


batch_size = 128
input_dim = 1024


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


def get_init_inputs():
    target = torch.randn(input_dim)
    return [target]