amari 0.9.4

# Dual Number GPU Integration Plan for v0.9.3

## Overview

This document outlines the GPU acceleration strategy for amari-dual automatic differentiation operations, building on the existing WebGPU/wgpu infrastructure in amari-gpu.

## Current Dual Number Infrastructure Analysis

### Existing Capabilities
- ✅ Single-variable dual numbers (`DualNumber<T>`)
- ✅ Multi-variable dual numbers (`MultiDualNumber<T>`)
- ✅ Complete mathematical function library (sin, cos, exp, etc.)
- ✅ Machine learning activation functions (ReLU, sigmoid, tanh)
- ✅ Chain rule automatic differentiation
- ✅ Comprehensive WASM bindings

### Performance Bottlenecks
- ❌ Sequential gradient computation for large parameter sets
- ❌ CPU-only Jacobian matrix computation
- ❌ No batch processing for automatic differentiation
- ❌ Limited scalability for large neural networks
- ❌ No GPU acceleration for tensor operations

## GPU Acceleration Opportunities

### High Priority (Maximum Impact)

#### 1. Batch Automatic Differentiation
**Current State**: Sequential AD computation on CPU
**GPU Benefits**: Massive parallelization for parameter gradients
**Use Cases**:
- Neural network training with thousands/millions of parameters
- Batch gradient computation for optimization algorithms
- Monte Carlo gradient estimation

**Implementation Strategy**:
```rust
pub async fn batch_forward_reverse_ad(
    &self,
    functions: &[Function],
    inputs: &[Vec<f64>],
    variables: &[usize],
) -> Result<Vec<(f64, Vec<f64>)>, GpuError>
```

#### 2. Jacobian Matrix Computation
**Current State**: Sequential computation of partial derivatives
**GPU Benefits**: Parallel computation of all partial derivatives
**Use Cases**:
- Sensitivity analysis for large systems
- Newton-Raphson optimization methods
- Machine learning gradient analysis

**Implementation Strategy**:
```rust
pub async fn compute_jacobian_batch(
    &self,
    vector_functions: &[VectorFunction],
    input_points: &[Vec<f64>],
) -> Result<Vec<Matrix<f64>>, GpuError>
```

#### 3. Neural Network Forward/Backward Pass
**Current State**: CPU-only layer-by-layer computation
**GPU Benefits**: Parallel layer computation with automatic gradients
**Use Cases**:
- Deep learning training acceleration
- Real-time neural network inference with gradients
- Neural architecture search

### Medium Priority (Significant Benefits)

#### 4. Hessian Matrix Computation
**Current State**: Not implemented (would require nested dual numbers)
**GPU Benefits**: Second-order optimization algorithms
**Use Cases**:
- Newton's method optimization
- Uncertainty quantification
- Curvature analysis

#### 5. Tensor Automatic Differentiation
**Current State**: Limited to scalar/vector operations
**GPU Benefits**: Full tensor AD for deep learning
**Use Cases**:
- Convolutional neural networks
- Tensor decomposition with gradients
- Advanced ML model training

#### 6. Stochastic Gradient Computation
**Current State**: Not implemented
**GPU Benefits**: Monte Carlo gradient estimation
**Use Cases**:
- Bayesian neural networks
- Stochastic optimization
- Uncertainty propagation

## Implementation Architecture

### 1. Core GPU Dual Number Infrastructure

```rust
/// GPU-accelerated automatic differentiation operations
pub struct GpuAutoDiff {
    device: wgpu::Device,
    queue: wgpu::Queue,

    // Compute pipelines for different AD operations
    forward_ad_pipeline: wgpu::ComputePipeline,
    reverse_ad_pipeline: wgpu::ComputePipeline,
    jacobian_pipeline: wgpu::ComputePipeline,
    hessian_pipeline: wgpu::ComputePipeline,
    neural_network_pipeline: wgpu::ComputePipeline,
}

impl GpuAutoDiff {
    pub async fn new() -> Result<Self, GpuError>;

    // Batch automatic differentiation
    pub async fn batch_forward_ad(...) -> Result<Vec<(f64, Vec<f64>)>, GpuError>;
    pub async fn batch_reverse_ad(...) -> Result<Vec<(f64, Vec<f64>)>, GpuError>;

    // Matrix computations
    pub async fn compute_jacobian_batch(...) -> Result<Vec<Matrix<f64>>, GpuError>;
    pub async fn compute_hessian_batch(...) -> Result<Vec<Matrix<f64>>, GpuError>;

    // Neural network operations
    pub async fn neural_network_forward_backward(...) -> Result<NetworkGradients, GpuError>;
    pub async fn batch_layer_gradients(...) -> Result<Vec<LayerGradients>, GpuError>;

    // Optimization support
    pub async fn stochastic_gradient_batch(...) -> Result<Vec<f64>, GpuError>;
    pub async fn gradient_descent_step_batch(...) -> Result<Vec<Vec<f64>>, GpuError>;
}
```

### 2. GPU-Optimized Dual Number Representation

```rust
/// GPU-compatible dual number representation
#[repr(C)]
#[derive(Copy, Clone, Pod, Zeroable)]
struct GpuDualNumber {
    value: f32,
    gradient: f32,
}

/// GPU-compatible multi-dual number (fixed-size for GPU efficiency)
#[repr(C)]
#[derive(Copy, Clone, Pod, Zeroable)]
struct GpuMultiDual<const N: usize> {
    value: f32,
    gradient: [f32; N],
}

/// Batch operations for efficient GPU computation
#[repr(C)]
#[derive(Copy, Clone, Pod, Zeroable)]
struct GpuDualBatch {
    batch_size: u32,
    parameter_count: u32,
    function_count: u32,
    _padding: u32,
}
```

### 3. WGSL Compute Shaders for Automatic Differentiation

#### Forward Mode Automatic Differentiation
```wgsl
// Forward mode AD for batch computation
struct DualNumber {
    value: f32,
    gradient: f32,
}

@group(0) @binding(0)
var<storage, read> input_duals: array<DualNumber>;

@group(0) @binding(1)
var<storage, read> function_codes: array<u32>; // Encoded function operations

@group(0) @binding(2)
var<storage, read_write> output_duals: array<DualNumber>;

// Dual number arithmetic operations
fn dual_add(a: DualNumber, b: DualNumber) -> DualNumber {
    return DualNumber(a.value + b.value, a.gradient + b.gradient);
}

fn dual_mul(a: DualNumber, b: DualNumber) -> DualNumber {
    return DualNumber(
        a.value * b.value,
        a.gradient * b.value + a.value * b.gradient
    );
}

fn dual_sin(a: DualNumber) -> DualNumber {
    return DualNumber(sin(a.value), a.gradient * cos(a.value));
}

fn dual_exp(a: DualNumber) -> DualNumber {
    let exp_val = exp(a.value);
    return DualNumber(exp_val, a.gradient * exp_val);
}

@compute @workgroup_size(64)
fn forward_ad_batch(@builtin(global_invocation_id) global_id: vec3<u32>) {
    let idx = global_id.x;
    if (idx >= arrayLength(&input_duals)) {
        return;
    }

    var current = input_duals[idx];

    // Execute function operations encoded in function_codes
    for (var i = 0u; i < arrayLength(&function_codes); i++) {
        let op_code = function_codes[i];

        switch op_code {
            case 1u: { // Addition
                current = dual_add(current, input_duals[idx + 1u]);
            }
            case 2u: { // Multiplication
                current = dual_mul(current, input_duals[idx + 1u]);
            }
            case 3u: { // Sine
                current = dual_sin(current);
            }
            case 4u: { // Exponential
                current = dual_exp(current);
            }
            default: {
                break;
            }
        }
    }

    output_duals[idx] = current;
}
```

#### Jacobian Matrix Computation
```wgsl
@group(0) @binding(0)
var<storage, read> input_batch: array<f32>; // Flattened input vectors

@group(0) @binding(1)
var<storage, read> function_batch: array<u32>; // Function definitions

@group(0) @binding(2)
var<storage, read_write> jacobian_batch: array<f32>; // Flattened Jacobian matrices

@compute @workgroup_size(16, 16)
fn compute_jacobian(@builtin(global_invocation_id) global_id: vec3<u32>) {
    let batch_idx = global_id.z;
    let output_idx = global_id.y;
    let input_idx = global_id.x;

    if (batch_idx >= batch_size || output_idx >= num_outputs || input_idx >= num_inputs) {
        return;
    }

    // Create dual number with unit gradient for current input variable
    var dual_input = DualNumber(
        input_batch[batch_idx * num_inputs + input_idx],
        select(0.0, 1.0, input_idx == global_id.x)
    );

    // Execute function to compute partial derivative
    let result = evaluate_function(dual_input, function_batch, output_idx);

    // Store partial derivative in Jacobian matrix
    let jacobian_offset = batch_idx * num_outputs * num_inputs;
    jacobian_batch[jacobian_offset + output_idx * num_inputs + input_idx] = result.gradient;
}
```

#### Neural Network Forward/Backward Pass
```wgsl
struct LayerParams {
    weights_offset: u32,
    bias_offset: u32,
    input_size: u32,
    output_size: u32,
    activation: u32, // 0=linear, 1=relu, 2=sigmoid, 3=tanh
}

@group(0) @binding(0)
var<storage, read> network_params: array<LayerParams>;

@group(0) @binding(1)
var<storage, read> weights_biases: array<f32>;

@group(0) @binding(2)
var<storage, read> input_batch: array<f32>;

@group(0) @binding(3)
var<storage, read_write> activations: array<f32>;

@group(0) @binding(4)
var<storage, read_write> gradients: array<f32>;

fn activate_dual(x: DualNumber, activation: u32) -> DualNumber {
    switch activation {
        case 1u: { // ReLU
            if (x.value > 0.0) {
                return x;
            } else {
                return DualNumber(0.0, 0.0);
            }
        }
        case 2u: { // Sigmoid
            let sig = 1.0 / (1.0 + exp(-x.value));
            return DualNumber(sig, x.gradient * sig * (1.0 - sig));
        }
        case 3u: { // Tanh
            let tanh_val = tanh(x.value);
            return DualNumber(tanh_val, x.gradient * (1.0 - tanh_val * tanh_val));
        }
        default: { // Linear
            return x;
        }
    }
}

@compute @workgroup_size(64)
fn neural_network_forward_backward(@builtin(global_invocation_id) global_id: vec3<u32>) {
    let batch_idx = global_id.x;
    let neuron_idx = global_id.y;
    let layer_idx = global_id.z;

    if (batch_idx >= batch_size || layer_idx >= num_layers) {
        return;
    }

    let layer = network_params[layer_idx];

    if (neuron_idx >= layer.output_size) {
        return;
    }

    // Compute weighted sum with automatic differentiation
    var weighted_sum = DualNumber(
        weights_biases[layer.bias_offset + neuron_idx],
        0.0
    );

    for (var i = 0u; i < layer.input_size; i++) {
        let input_val = if (layer_idx == 0u) {
            input_batch[batch_idx * layer.input_size + i]
        } else {
            activations[(layer_idx - 1u) * batch_size * max_layer_size + batch_idx * max_layer_size + i]
        };

        let weight = weights_biases[layer.weights_offset + neuron_idx * layer.input_size + i];
        let input_dual = DualNumber(input_val, select(0.0, 1.0, i == target_input_idx));
        let weight_dual = DualNumber(weight, select(0.0, 1.0, is_weight_target));

        weighted_sum = dual_add(weighted_sum, dual_mul(input_dual, weight_dual));
    }

    // Apply activation function
    let activated = activate_dual(weighted_sum, layer.activation);

    // Store forward pass result
    activations[layer_idx * batch_size * max_layer_size + batch_idx * max_layer_size + neuron_idx] = activated.value;

    // Store gradient for backward pass
    gradients[layer_idx * batch_size * max_layer_size + batch_idx * max_layer_size + neuron_idx] = activated.gradient;
}
```

### 4. Adaptive Dispatch Strategy

```rust
impl GpuAutoDiff {
    /// Determine optimal computation strategy based on problem size
    pub fn should_use_gpu_for_ad(operation: ADOperation, problem_size: ADProblemSize) -> bool {
        match operation {
            ADOperation::ForwardAD => {
                problem_size.num_variables >= 100 && problem_size.batch_size >= 32
            }
            ADOperation::ReverseAD => {
                problem_size.num_outputs <= 10 && problem_size.num_variables >= 1000
            }
            ADOperation::JacobianComputation => {
                problem_size.num_variables * problem_size.num_outputs >= 10000
            }
            ADOperation::NeuralNetworkTraining => {
                problem_size.total_parameters >= 1000 && problem_size.batch_size >= 16
            }
        }
    }
}

#[derive(Debug, Clone)]
pub struct ADProblemSize {
    pub num_variables: usize,
    pub num_outputs: usize,
    pub batch_size: usize,
    pub total_parameters: usize,
    pub function_complexity: usize,
}
```

### 5. Integration with Machine Learning Frameworks

```rust
/// High-level ML integration for GPU-accelerated automatic differentiation
pub struct GpuNeuralNetwork {
    gpu_autodiff: GpuAutoDiff,
    layers: Vec<LayerDefinition>,
    parameters: Vec<f32>,
}

impl GpuNeuralNetwork {
    pub async fn forward_with_gradients(
        &self,
        inputs: &[Vec<f32>],
        target_parameters: &[usize],
    ) -> Result<(Vec<f32>, Vec<f32>), GpuError>;

    pub async fn backward_pass(
        &self,
        loss_gradients: &[f32],
    ) -> Result<Vec<f32>, GpuError>;

    pub async fn gradient_descent_step(
        &mut self,
        learning_rate: f32,
        gradients: &[f32],
    ) -> Result<(), GpuError>;
}
```

## Performance Targets

### Benchmarking Goals
- **Batch Forward AD**: 10-100x speedup for >1000 variables
- **Jacobian Computation**: 20-200x speedup for large matrices (>100x100)
- **Neural Network Training**: 5-50x speedup for networks >1000 parameters
- **Memory Efficiency**: <3x GPU memory overhead vs minimal CPU implementation

### Scalability Targets
- Support for up to 1M parameters in batch AD
- Jacobian matrices up to 10,000 x 10,000
- Neural networks with >100 layers and >1M parameters
- Batch sizes up to 1024 for training

## Implementation Timeline

### Phase 1: Core Infrastructure (Week 1)
- [ ] Extend amari-gpu with automatic differentiation support
- [ ] Implement `GpuAutoDiff` base structure
- [ ] Create batch forward-mode AD GPU pipeline
- [ ] Add adaptive dispatch for AD operations

### Phase 2: Jacobian Acceleration (Week 2)
- [ ] Implement GPU Jacobian matrix computation
- [ ] Add batch processing for multiple Jacobian computations
- [ ] Optimize memory usage for large matrices
- [ ] Integration with existing amari-dual types

### Phase 3: Neural Network Integration (Week 3)
- [ ] GPU neural network forward/backward pass
- [ ] Batch gradient computation for multiple networks
- [ ] Layer-wise gradient accumulation and optimization
- [ ] Integration with popular ML patterns

### Phase 4: Advanced Operations (Week 4)
- [ ] Hessian matrix computation
- [ ] Second-order optimization algorithms
- [ ] Stochastic gradient computation
- [ ] Comprehensive benchmarking and optimization

## Testing Strategy

### Correctness Verification
- Gradient checking against finite differences
- Cross-validation with existing CPU AD implementations
- Mathematical property verification (chain rule, linearity)
- Precision analysis for f32 vs f64 computations

### Performance Validation
- Scaling benchmarks across different problem sizes
- Memory usage profiling and optimization
- Comparative analysis vs leading AD frameworks
- Real-world machine learning workload testing

### Robustness Testing
- Edge case handling (zero gradients, numerical instabilities)
- Large-scale stress testing (millions of parameters)
- Memory pressure testing
- Cross-platform GPU compatibility

## Risk Mitigation

### Technical Challenges
1. **Numerical Precision**: Implement mixed-precision strategies and validation
2. **GPU Memory Constraints**: Design streaming and chunked computation algorithms
3. **Complex Function Support**: Modular shader design for extensible operations
4. **Debuggability**: Comprehensive logging and gradient verification tools

### Performance Risks
1. **Small Problem Overhead**: Careful threshold tuning for GPU dispatch
2. **Memory Bandwidth**: Optimize data layout and minimize transfers
3. **Divergent Execution**: Design GPU-friendly control flow patterns

## Success Metrics

### Quantitative Goals
- [ ] 10x+ speedup for large-scale automatic differentiation
- [ ] 20x+ speedup for Jacobian matrix computation
- [ ] 5x+ speedup for neural network training
- [ ] <20% GPU memory overhead for typical workloads

### Integration Goals
- [ ] Seamless integration with existing amari-dual API
- [ ] Backward compatibility with all existing AD operations
- [ ] Clear performance characteristics and usage guidelines
- [ ] Comprehensive error handling and fallback mechanisms

## Future Extensions

### Advanced Automatic Differentiation
- Higher-order derivatives (Hessians, tensor derivatives)
- Sparse automatic differentiation for structured problems
- Checkpointing strategies for memory-efficient reverse-mode AD

### Distributed Computation
- Multi-GPU automatic differentiation
- Distributed gradient computation across nodes
- Integration with distributed training frameworks

### Specialized Applications
- GPU-accelerated Bayesian neural networks
- Automatic differentiation for scientific computing
- Real-time optimization with automatic gradients

---

This plan provides a comprehensive roadmap for integrating automatic differentiation operations with GPU acceleration, enabling high-performance machine learning and optimization applications while maintaining the mathematical rigor and correctness of the amari-dual system.