Kolmogorov-Arnold Networks: Complete Implementation Guide

Kolmogorov-Arnold Networks: Complete Implementation Guide

Introduction

Kolmogorov-Arnold Networks (KANs) represent a paradigm shift in neural network architecture, drawing inspiration from the mathematical foundations laid by Andrey Kolmogorov and Vladimir Arnold in the 1950s. Unlike traditional Multi-Layer Perceptrons (MLPs) that place learnable parameters on nodes, KANs position learnable activation functions on edges, fundamentally changing how neural networks process and learn from data.

Architecture Overview

Traditional MLPs vs KANs

Multi-Layer Perceptrons (MLPs): - Learnable parameters: weights and biases on nodes - Fixed activation functions (ReLU, sigmoid, etc.) - Linear transformations followed by pointwise nonlinearities

Kolmogorov-Arnold Networks (KANs): - Learnable parameters: activation functions on edges - No traditional weight matrices - Each edge has its own learnable univariate function

Mathematical Formulation

Layer-wise Computation

For a KAN with L layers, the computation at layer l can be expressed as:

# Pseudocode for KAN layer computation
def kan_layer_forward(x, phi_functions):
    """
    x: input tensor of shape (batch_size, input_dim)
    phi_functions: learnable univariate functions for each edge
    """
    output = torch.zeros(batch_size, output_dim)
    
    for i in range(input_dim):
        for j in range(output_dim):
            # Apply learnable activation function φ_{i,j} to input x_i
            output[:, j] += phi_functions[i][j](x[:, i])
    
    return output

Learnable Activation Functions

The core innovation of KANs lies in the learnable activation functions. These are typically implemented using:

1. B-splines: Piecewise polynomial functions that provide smooth, differentiable approximations
2. Residual connections: Allow the network to learn both the spline component and a base function
3. Grid-based parameterization: Enables efficient computation and gradient flow

Implementation Details

B-spline Based Activation Functions

import torch
import torch.nn as nn
import torch.nn.functional as F
import math

class BSplineActivation(nn.Module):
    def __init__(self, grid_size=5, spline_order=3, grid_range=(-1, 1)):
        super().__init__()
        self.grid_size = grid_size
        self.spline_order = spline_order
        self.grid_range = grid_range
        
        # Create uniform grid
        self.register_buffer('grid', torch.linspace(
            grid_range[0], grid_range[1], grid_size + 1
        ))
        
        # Extend grid for B-spline computation
        h = (grid_range[1] - grid_range[0]) / grid_size
        extended_grid = torch.cat([
            torch.arange(grid_range[0] - spline_order * h, grid_range[0], h),
            self.grid,
            torch.arange(grid_range[1] + h, grid_range[1] + (spline_order + 1) * h, h)
        ])
        self.register_buffer('extended_grid', extended_grid)
        
        # Learnable coefficients for B-spline
        self.coefficients = nn.Parameter(
            torch.randn(grid_size + spline_order)
        )
        
        # Scale parameter for the activation
        self.scale = nn.Parameter(torch.ones(1))
        
    def forward(self, x):
        # Compute B-spline basis functions
        batch_size = x.shape[0]
        x_expanded = x.unsqueeze(-1)  # (batch_size, 1)
        
        # Compute B-spline values
        spline_values = self.compute_bspline(x_expanded)
        
        # Linear combination with learnable coefficients
        output = torch.sum(spline_values * self.coefficients, dim=-1)
        
        return self.scale * output
    
    def compute_bspline(self, x):
        """Compute B-spline basis functions using Cox-de Boor recursion"""
        grid = self.extended_grid
        order = self.spline_order
        
        # Initialize basis functions
        basis = torch.zeros(x.shape[0], len(grid) - 1, device=x.device)
        
        # Find intervals
        for i in range(len(grid) - 1):
            mask = (x.squeeze(-1) >= grid[i]) & (x.squeeze(-1) < grid[i + 1])
            basis[mask, i] = 1.0
        
        # Cox-de Boor recursion
        for k in range(1, order + 1):
            new_basis = torch.zeros_like(basis)
            for i in range(len(grid) - k - 1):
                if grid[i + k] != grid[i]:
                    alpha1 = (x.squeeze(-1) - grid[i]) / (grid[i + k] - grid[i])
                    new_basis[:, i] += alpha1 * basis[:, i]
                
                if grid[i + k + 1] != grid[i + 1]:
                    alpha2 = (grid[i + k + 1] - x.squeeze(-1)) / (grid[i + k + 1] - grid[i + 1])
                    new_basis[:, i] += alpha2 * basis[:, i + 1]
            
            basis = new_basis
        
        return basis[:, :len(self.coefficients)]

class KANLayer(nn.Module):
    def __init__(self, input_dim, output_dim, grid_size=5, spline_order=3):
        super().__init__()
        self.input_dim = input_dim
        self.output_dim = output_dim
        
        # Create learnable activation functions for each edge
        self.activations = nn.ModuleList([
            nn.ModuleList([
                BSplineActivation(grid_size, spline_order) 
                for _ in range(output_dim)
            ]) for _ in range(input_dim)
        ])
        
        # Base linear transformation (residual connection)
        self.base_weight = nn.Parameter(torch.randn(input_dim, output_dim) * 0.1)
        
    def forward(self, x):
        batch_size = x.shape[0]
        output = torch.zeros(batch_size, self.output_dim, device=x.device)
        
        # Apply learnable activations
        for i in range(self.input_dim):
            for j in range(self.output_dim):
                activated = self.activations[i][j](x[:, i])
                output[:, j] += activated
        
        # Add base linear transformation
        base_output = torch.matmul(x, self.base_weight)
        
        return output + base_output

class KolmogorovArnoldNetwork(nn.Module):
    def __init__(self, layer_dims, grid_size=5, spline_order=3):
        super().__init__()
        self.layers = nn.ModuleList()
        
        for i in range(len(layer_dims) - 1):
            layer = KANLayer(
                layer_dims[i], 
                layer_dims[i + 1], 
                grid_size, 
                spline_order
            )
            self.layers.append(layer)
    
    def forward(self, x):
        for layer in self.layers:
            x = layer(x)
        return x
    
    def regularization_loss(self, regularization_factor=1e-4):
        """Compute regularization loss to encourage sparsity"""
        reg_loss = 0
        for layer in self.layers:
            for i in range(layer.input_dim):
                for j in range(layer.output_dim):
                    # L1 regularization on activation function coefficients
                    reg_loss += torch.sum(torch.abs(layer.activations[i][j].coefficients))
        
        return regularization_factor * reg_loss

Training Loop Implementation

def train_kan(model, train_loader, val_loader, epochs=100, lr=1e-3):
    optimizer = torch.optim.AdamW(model.parameters(), lr=lr, weight_decay=1e-4)
    scheduler = torch.optim.lr_scheduler.ReduceLROnPlateau(
        optimizer, mode='min', factor=0.5, patience=10
    )
    
    criterion = nn.MSELoss()
    
    train_losses = []
    val_losses = []
    
    for epoch in range(epochs):
        # Training phase
        model.train()
        train_loss = 0
        for batch_idx, (data, target) in enumerate(train_loader):
            optimizer.zero_grad()
            
            output = model(data)
            loss = criterion(output, target)
            
            # Add regularization
            reg_loss = model.regularization_loss()
            total_loss = loss + reg_loss
            
            total_loss.backward()
            
            # Gradient clipping for stability
            torch.nn.utils.clip_grad_norm_(model.parameters(), max_norm=1.0)
            
            optimizer.step()
            train_loss += total_loss.item()
        
        # Validation phase
        model.eval()
        val_loss = 0
        with torch.no_grad():
            for data, target in val_loader:
                output = model(data)
                val_loss += criterion(output, target).item()
        
        avg_train_loss = train_loss / len(train_loader)
        avg_val_loss = val_loss / len(val_loader)
        
        train_losses.append(avg_train_loss)
        val_losses.append(avg_val_loss)
        
        scheduler.step(avg_val_loss)
        
        if epoch % 10 == 0:
            print(f'Epoch {epoch}: Train Loss = {avg_train_loss:.6f}, '
                  f'Val Loss = {avg_val_loss:.6f}')
    
    return train_losses, val_losses

Advanced Features and Optimizations

1. Pruning and Sparsification

def prune_kan(model, threshold=1e-2):
    """Remove edges with small activation function magnitudes"""
    with torch.no_grad():
        for layer in model.layers:
            for i in range(layer.input_dim):
                for j in range(layer.output_dim):
                    activation = layer.activations[i][j]
                    
                    # Compute magnitude of activation function
                    magnitude = torch.norm(activation.coefficients)
                    
                    if magnitude < threshold:
                        # Zero out the activation function
                        activation.coefficients.fill_(0)
                        activation.scale.fill_(0)

2. Symbolic Regression Capabilities

def symbolic_extraction(model, input_names, output_names):
    """Extract symbolic expressions from trained KAN"""
    expressions = []
    
    for layer_idx, layer in enumerate(model.layers):
        layer_expressions = []
        
        for j in range(layer.output_dim):
            terms = []
            
            for i in range(layer.input_dim):
                activation = layer.activations[i][j]
                
                # Check if activation is significant
                if torch.norm(activation.coefficients) > 1e-3:
                    # Fit simple function to activation
                    func_type = fit_symbolic_function(activation)
                    terms.append(f"{func_type}({input_names[i]})")
            
            if terms:
                expression = " + ".join(terms)
                layer_expressions.append(expression)
        
        expressions.append(layer_expressions)
    
    return expressions

def fit_symbolic_function(activation):
    """Fit symbolic function to learned activation"""
    # Sample the activation function
    x_test = torch.linspace(-1, 1, 100)
    y_test = activation(x_test).detach()
    
    # Try fitting common functions
    functions = {
        'linear': lambda x, a, b: a * x + b,
        'quadratic': lambda x, a, b, c: a * x**2 + b * x + c,
        'sin': lambda x, a, b, c: a * torch.sin(b * x + c),
        'exp': lambda x, a, b: a * torch.exp(b * x),
        'tanh': lambda x, a, b: a * torch.tanh(b * x)
    }
    
    best_fit = 'linear'  # Default
    min_error = float('inf')
    
    for func_name, func in functions.items():
        try:
            # Simplified fitting (in practice, use scipy.optimize)
            if func_name == 'linear':
                # Simple linear regression
                A = torch.stack([x_test, torch.ones_like(x_test)], dim=1)
                params = torch.linalg.lstsq(A, y_test).solution
                pred = func(x_test, params[0], params[1])
            else:
                # Use first-order approximation
                pred = y_test  # Placeholder
            
            error = torch.mean((y_test - pred)**2)
            
            if error < min_error:
                min_error = error
                best_fit = func_name
        
        except:
            continue
    
    return best_fit

3. Grid Adaptation

def adaptive_grid_refinement(model, train_loader, refinement_factor=2):
    """Adapt grid points based on function complexity"""
    model.eval()
    
    with torch.no_grad():
        # Collect statistics on activation function usage
        activation_stats = {}
        
        for batch_idx, (data, target) in enumerate(train_loader):
            if batch_idx > 10:  # Sample a few batches
                break
                
            for layer_idx, layer in enumerate(model.layers):
                if layer_idx not in activation_stats:
                    activation_stats[layer_idx] = {}
                
                for i in range(layer.input_dim):
                    for j in range(layer.output_dim):
                        key = (i, j)
                        if key not in activation_stats[layer_idx]:
                            activation_stats[layer_idx][key] = []
                        
                        # Record input values for this activation
                        if layer_idx == 0:
                            input_vals = data[:, i]
                        else:
                            # Would need to track intermediate activations
                            input_vals = data[:, i]  # Simplified
                        
                        activation_stats[layer_idx][key].extend(
                            input_vals.cpu().numpy().tolist()
                        )
        
        # Refine grids based on usage patterns
        for layer_idx, layer in enumerate(model.layers):
            for i in range(layer.input_dim):
                for j in range(layer.output_dim):
                    activation = layer.activations[i][j]
                    key = (i, j)
                    
                    if key in activation_stats[layer_idx]:
                        input_range = activation_stats[layer_idx][key]
                        
                        # Compute density and refine grid
                        hist, bins = torch.histogram(
                            torch.tensor(input_range), bins=activation.grid_size
                        )
                        
                        # Areas with high density get more grid points
                        high_density_regions = hist > hist.mean()
                        
                        if high_density_regions.any():
                            # Refine grid (simplified implementation)
                            new_grid_size = activation.grid_size * refinement_factor
                            # Would need to properly interpolate coefficients

Practical Applications and Use Cases

1. Function Approximation

# Example: Approximating a complex mathematical function
def test_function_approximation():
    # Generate synthetic data
    def target_function(x):
        return torch.sin(x[:, 0]) * torch.cos(x[:, 1]) + 0.5 * x[:, 0]**2
    
    # Create dataset
    n_samples = 1000
    x = torch.randn(n_samples, 2)
    y = target_function(x).unsqueeze(1)
    
    # Split data
    train_size = int(0.8 * n_samples)
    train_x, test_x = x[:train_size], x[train_size:]
    train_y, test_y = y[:train_size], y[train_size:]
    
    # Create KAN model
    model = KolmogorovArnoldNetwork([2, 5, 1], grid_size=10)
    
    # Train model
    train_dataset = torch.utils.data.TensorDataset(train_x, train_y)
    train_loader = torch.utils.data.DataLoader(train_dataset, batch_size=32)
    
    val_dataset = torch.utils.data.TensorDataset(test_x, test_y)
    val_loader = torch.utils.data.DataLoader(val_dataset, batch_size=32)
    
    train_losses, val_losses = train_kan(model, train_loader, val_loader)
    
    # Evaluate
    model.eval()
    with torch.no_grad():
        predictions = model(test_x)
        mse = torch.mean((predictions - test_y)**2)
        print(f"Test MSE: {mse.item():.6f}")
    
    return model, train_losses, val_losses

2. Scientific Computing

# Example: Solving differential equations
def solve_pde_with_kan():
    """Use KAN to solve partial differential equations"""
    
    class PDESolver(nn.Module):
        def __init__(self):
            super().__init__()
            self.kan = KolmogorovArnoldNetwork([2, 10, 10, 1])
        
        def forward(self, x, t):
            inputs = torch.stack([x, t], dim=1)
            return self.kan(inputs)
        
        def physics_loss(self, x, t):
            """Compute physics-informed loss for PDE"""
            x.requires_grad_(True)
            t.requires_grad_(True)
            
            u = self.forward(x, t)
            
            # Compute derivatives
            u_t = torch.autograd.grad(
                u, t, torch.ones_like(u), create_graph=True
            )[0]
            
            u_x = torch.autograd.grad(
                u, x, torch.ones_like(u), create_graph=True
            )[0]
            
            u_xx = torch.autograd.grad(
                u_x, x, torch.ones_like(u_x), create_graph=True
            )[0]
            
            # PDE residual: u_t - u_xx = 0 (heat equation)
            pde_residual = u_t - u_xx
            
            return torch.mean(pde_residual**2)
    
    # Training would involve minimizing physics loss
    # along with boundary and initial conditions
    return PDESolver()

Performance Analysis and Comparisons

Computational Complexity

Memory Complexity: - MLPs: O(Σ(n_i × n_{i+1})) where n_i is the number of neurons in layer i - KANs: O(Σ(n_i × n_{i+1} × G)) where G is the grid size for B-splines

Time Complexity: - Forward pass: O(Σ(n_i × n_{i+1} × G × k)) where k is the spline order - Backward pass: Similar, with additional complexity for B-spline derivative computation

Advantages of KANs

1. Interpretability: Learnable activation functions can be visualized and analyzed
2. Expressiveness: Can represent complex functions with fewer parameters in some cases
3. Scientific Computing: Natural fit for problems requiring symbolic regression
4. Adaptive Capacity: Can learn specialized activation functions for different parts of the input space

Limitations

1. Computational Overhead: B-spline computation is more expensive than simple activations
2. Memory Usage: Requires more memory due to grid-based parameterization
3. Training Stability: Can be more sensitive to hyperparameter choices
4. Limited Scale: Current implementations don’t scale to very large networks easily

Best Practices and Hyperparameter Tuning

Grid Size Selection

def tune_grid_size(data_complexity, input_dim):
    """Heuristic for selecting appropriate grid size"""
    base_grid_size = max(5, min(20, int(math.log(data_complexity) * 2)))
    
    # Adjust based on input dimensionality
    if input_dim > 10:
        base_grid_size = max(3, base_grid_size - 2)
    elif input_dim < 3:
        base_grid_size = min(25, base_grid_size + 3)
    
    return base_grid_size

Regularization Strategies

def advanced_regularization(model, l1_factor=1e-4, smoothness_factor=1e-3):
    """Comprehensive regularization for KANs"""
    reg_loss = 0
    
    for layer in model.layers:
        for i in range(layer.input_dim):
            for j in range(layer.output_dim):
                activation = layer.activations[i][j]
                
                # L1 regularization for sparsity
                l1_loss = torch.sum(torch.abs(activation.coefficients))
                
                # Smoothness regularization
                if len(activation.coefficients) > 1:
                    smoothness_loss = torch.sum(
                        (activation.coefficients[1:] - activation.coefficients[:-1])**2
                    )
                else:
                    smoothness_loss = 0
                
                reg_loss += l1_factor * l1_loss + smoothness_factor * smoothness_loss
    
    return reg_loss

Future Directions and Research Opportunities

1. Scalability Improvements

• Efficient GPU implementations of B-spline computations
• Sparse KAN architectures for high-dimensional problems
• Distributed training strategies

2. Theoretical Developments

• Approximation theory for KAN architectures
• Convergence guarantees and optimization landscapes
• Connections to other function approximation methods

3. Application Domains

• Scientific machine learning and physics-informed neural networks
• Automated theorem proving and symbolic computation
• Interpretable AI for critical applications

Conclusion

Kolmogorov-Arnold Networks represent a fundamental rethinking of neural network architecture, moving from node-based to edge-based learnable parameters. While they face challenges in terms of computational efficiency and scalability, their unique properties make them particularly well-suited for scientific computing, interpretable AI, and function approximation tasks.

The mathematical elegance of KANs, rooted in classical approximation theory, combined with their practical capabilities for symbolic regression and interpretable modeling, positions them as an important tool in the modern machine learning toolkit. As implementation techniques improve and computational bottlenecks are addressed, we can expect to see broader adoption of KAN-based approaches across various domains.

The code implementations provided here offer a foundation for experimenting with KANs, but ongoing research continues to refine these architectures and explore their full potential. Whether KANs will revolutionize neural network design remains to be seen, but they certainly offer a compelling alternative perspective on how neural networks can learn and represent complex functions.