GR-KAN explanation: "**Summary of GR-KAN (Group-Rational KAN):**

GR-KAN is an advanced variation of the Kolmogorov-Arnold Network (KAN), introduced as a solution for integrating KANs into transformers like the Kolmogorov–Arnold Transformer (KAT). GR-KAN addresses three major challenges in scaling KAN-based architectures (computational inefficiency, parameter overhead, and initialization issues) by employing group-wise parameter sharing and rational functions to replace the standard MLP layers in transformers. This improves computational efficiency, scalability, and performance for tasks like image recognition and object detection.

### Detailed Description of the Three Solutions

1. **Rational Activation Functions (S1)**:
   - **Challenge (C1)**: The original B-spline functions used in KAN layers were inefficient for parallel computation, especially on modern GPUs. B-splines required recursive calculations, leading to slow processing.
   - **Solution (S1)**: The B-spline functions were replaced with *rational functions*, which are a more GPU-friendly mathematical operation. Rational functions consist of ratios of polynomials, offering both flexibility and computational efficiency. They are better suited for high-performance GPU environments because polynomials involve simpler operations that can be executed in parallel.
     - The rational function \( F(x) \) is expressed as \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and these are optimized using CUDA to ensure efficient gradient computation during training.
     - To prevent instability caused by poles (where the denominator \( Q(x) \) approaches zero), a modification called the Safe Padé Activation Unit (PAU) was introduced, ensuring that the model remains numerically stable during training.
     - By using rational functions, GR-KAN improves upon the original KAN by making these activation functions significantly faster on GPUs, reducing the FLOPs (floating point operations per second) required for each evaluation.

2. **Group KAN (S2)**:
   - **Challenge (C2)**: KAN's architecture required a unique base function for each input-output pair, leading to excessive parameter growth and computational overhead as network sizes increased. This made it difficult to scale KAN-based models for large datasets like ImageNet.
   - **Solution (S2)**: GR-KAN introduces a *group-wise parameter sharing* strategy. Instead of assigning individual functions for every input-output pair, GR-KAN shares function coefficients among groups of neurons. Specifically, input channels are divided into groups, and within each group, the same rational function is applied, significantly reducing the number of unique functions that need to be computed.
     - This grouping mechanism drastically cuts down on the number of parameters and computational requirements without compromising the model's ability to learn complex functions. 
     - For instance, if the number of input channels is \( d_{\text{in}} \) and the number of output channels is \( d_{\text{out}} \), a standard KAN would have required \( d_{\text{in}} \times d_{\text{out}} \) activation functions. GR-KAN reduces this by grouping the inputs and outputs, applying shared functions within each group, thereby reducing the parameter size and computational load.
     - This technique also enhances the model’s ability to scale, as it no longer faces the exponential parameter growth seen in standard KAN.

3. **Variance-Preserving Initialization (S3)**:
   - **Challenge (C3)**: KAN’s learnable activation functions were not properly initialized, leading to instability during training. The default initialization techniques did not maintain variance across layers, causing signal propagation issues.
   - **Solution (S3)**: GR-KAN addresses this by using a *variance-preserving initialization* strategy. This ensures that the variance of activations remains stable as they pass through the network, preventing vanishing or exploding gradients, which are common issues in deep learning.
     - The rational function's coefficients (for both the numerator and denominator) are initialized in such a way that the variance of the output is preserved across layers. Specifically, coefficients are initialized to mimic established activation functions like ReLU or GELU, which have known variance properties.
     - After the initial setup of the coefficients, GR-KAN adjusts the scale of the weights based on the variance ratio between the input and the expected output. This guarantees that the activations retain their variance, ensuring smooth convergence during training."

Relevant equations: "1. Generalized Kolmogorov–Arnold layer:
\[
f(x) = \Phi \circ x = \left[ \sum_{i=1}^{d_{in}} \phi_{i,1}(x_i) \dots \sum_{i=1}^{d_{in}} \phi_{i,d_{out}}(x_i) \right]
\]
where \( \Phi \) is a matrix defined as:
\[
\Phi = 
\begin{pmatrix}
\phi_{1,1}(\cdot) & \dots & \phi_{1,d_{in}}(\cdot) \\
\vdots & \ddots & \vdots \\
\phi_{d_{out},1}(\cdot) & \dots & \phi_{d_{out},d_{in}}(\cdot)
\end{pmatrix}
\]
This matrix form extends to a KAN network composed of stacked layers.

2. Group-Rational KAN (GR-KAN):
\[
\text{GR-KAN}(x) = \Phi \circ x = \left[ \sum_{i=1}^{d_{in}} w_{i,1} F_{\lfloor i / d_g \rfloor}(x_i) \dots \sum_{i=1}^{d_{in}} w_{i,d_{out}} F_{\lfloor i / d_g \rfloor}(x_i) \right]
\]
This can be simplified in matrix form:
\[
\text{GR-KAN}(x) = W F(x)
\]
where \( W \in \mathbb{R}^{d_{in} \times d_{out}} \) and \( F(x) \) is the group-wise rational function applied elementwise.

3. Rational function in GR-KAN:
\[
\phi(x) = w F(x) = w \frac{P(x)}{Q(x)} = w \frac{a_0 + a_1 x + \dots + a_m x^m}{b_0 + b_1 x + \dots + b_n x^n}
\]
This represents the rational activation function.

4. Horner’s method for efficient polynomial evaluation:
\[
a_0 + a_1 x + \dots + a_m x^m = a_0 + x(a_1 + x(a_2 + \dots))
\]

These are the core equations relevant to GR-KAN and KAN layers  ."

Based on the above, write GR-KAN layer code in pytorch.