Amari v0.9.8

Multi-GPU Mathematical Computing Platform with Intelligent Load Balancing

A comprehensive mathematical computing library featuring geometric algebra, relativistic physics, tropical algebra, automatic differentiation, and information geometry. The library provides complete multi-GPU infrastructure with intelligent workload distribution, advanced profiling systems, and optimization algorithms across all mathematical domains.

Features

Core Mathematical Systems

Geometric Algebra (Clifford Algebra): Multivectors, rotors, and geometric products for 3D rotations and spatial transformations
Relativistic Physics: Complete spacetime algebra (Cl(1,3)) with Minkowski signature for relativistic calculations
Tropical Algebra: Max-plus semiring operations for optimization and neural network applications
Automatic Differentiation: Forward-mode AD with dual numbers for exact derivatives
Fusion Systems: Tropical-dual-Clifford fusion combining three algebraic systems
Information Geometry: Statistical manifolds, KL/JS divergences, and Fisher information
Optimization: Gradient-based optimization with geometric constraints
Network Analysis: Geometric network analysis and graph neural networks
Cellular Automata: Geometric automata with configurable rules
Enumerative Geometry: Algebraic curves and enumerative computations

Multi-GPU Infrastructure

Multi-GPU Architecture: Infrastructure supporting up to 8 GPUs with intelligent workload distribution
Advanced Load Balancing: Five strategies including Balanced, CapabilityAware, MemoryAware, LatencyOptimized, and Adaptive
Performance Profiling: Timeline analysis with microsecond precision and automatic bottleneck detection
Comprehensive Benchmarking: Production-ready validation across all mathematical domains
Scaling Efficiency: Realistic performance modeling with 90%/80%/70% efficiency for 2/4/8 GPUs
Graceful Degradation: Automatic fallback to single GPU or CPU when multi-GPU unavailable

High-Precision Arithmetic

Spacecraft Orbital Mechanics: High-precision arithmetic for critical trajectory calculations with configurable tolerance
Geodesic Integration: Velocity Verlet method for curved spacetime particle trajectories
Schwarzschild Metric: Spherically symmetric gravitational fields for astrophysics applications
Phantom Types: Compile-time verification of relativistic invariants and spacetime signatures

Platform Support

Native Rust: High-performance execution with rug (GMP/MPFR) backend for high-precision arithmetic
WebAssembly: Full-featured WASM bindings with dashu backend for browser compatibility
Universal Precision: Consistent API and mathematical accuracy across all platforms
GPU Acceleration: WebGPU support for large-scale parallel computations
TypeScript Support: Complete TypeScript definitions included
Deployment Freedom: Pure Rust WASM builds deploy anywhere without system dependencies
Cross-Platform: Linux, macOS, Windows, browsers, Node.js, and edge computing environments

Installation

Rust Crates

Add to your Cargo.toml:

[dependencies]
# Core geometric algebra and mathematical foundations
amari-core = "0.9.8"

# High-precision relativistic physics with multi-backend support
amari-relativistic = { version = "0.9.8", features = ["high-precision"] }

# For native applications (uses rug/GMP backend)
amari-relativistic = { version = "0.9.8", features = ["native-precision"] }

# For WebAssembly targets (uses dashu backend)
amari-relativistic = { version = "0.9.8", features = ["wasm-precision"] }

# Multi-GPU acceleration and intelligent load balancing
amari-gpu = "0.9.8"

# Optimization algorithms
amari-optimization = "0.9.8"

# Additional mathematical systems
amari-tropical = "0.9.8"
amari-dual = "0.9.8"
amari-info-geom = "0.9.8"
amari-automata = "0.9.8"
amari-fusion = "0.9.8"
amari-network = "0.9.8"
amari-enumerative = "0.9.8"

JavaScript/TypeScript (WebAssembly)

npm install @justinelliottcobb/amari-wasm

Or with yarn:

yarn add @justinelliottcobb/amari-wasm

Quick Start

Rust: Spacecraft Orbital Mechanics

use amari_relativistic::prelude::*;

fn main() -> Result<(), Box<dyn std::error::Error>> {
    // Create gravitational field from massive object (Earth)
    let earth = schwarzschild::SchwarzschildMetric::earth();
    let mut integrator = geodesic::GeodesicIntegrator::with_metric(Box::new(earth));

    // Create spacecraft at 400 km altitude
    let altitude = 400e3; // 400 km
    let earth_radius = 6.371e6; // Earth radius
    let position = Vector3::new(earth_radius + altitude, 0.0, 0.0);
    let orbital_velocity = Vector3::new(0.0, 7.67e3, 0.0); // ~7.67 km/s orbital velocity

    // Create spacecraft particle
    let mut spacecraft = particle::RelativisticParticle::new(
        position,
        orbital_velocity,
        0.0, // Uncharged
        1000.0, // 1000 kg spacecraft
        0.0, // No charge
    )?;

    // Propagate orbit for one period with high precision
    let orbital_period = 5580.0; // ~93 minutes
    let trajectory = particle::propagate_relativistic(
        &mut spacecraft,
        &mut integrator,
        orbital_period,
        60.0, // 1-minute time steps
    )?;

    println!("Spacecraft orbital trajectory computed with {} points", trajectory.len());
    println!("Final position: {:?}", spacecraft.position_3d());

    Ok(())
}

Rust: Multi-GPU Load Balancing

use amari_gpu::{
    SharedGpuContext, IntelligentLoadBalancer, LoadBalancingStrategy,
    Workload, ComputeIntensity, BenchmarkRunner
};

async fn main() -> Result<(), Box<dyn std::error::Error>> {
    // Initialize multi-GPU context
    let context = SharedGpuContext::with_multi_gpu().await?;
    println!("Initialized {} GPU devices", context.device_count().await);

    // Create intelligent load balancer
    let load_balancer = IntelligentLoadBalancer::new(LoadBalancingStrategy::CapabilityAware);

    // Define computational workload
    let workload = Workload {
        operation_type: "geometric_product".to_string(),
        data_size: 100000,
        memory_requirement_mb: 256.0,
        compute_intensity: ComputeIntensity::Heavy,
        parallelizable: true,
        synchronization_required: true,
    };

    // Distribute workload across available GPUs
    let assignments = load_balancer.distribute_workload(&workload).await?;
    println!("Workload distributed across {} devices", assignments.len());

    for assignment in &assignments {
        println!(
            "Device {}: {:.1}% workload, estimated completion: {:.2}ms",
            assignment.device_id.0,
            assignment.workload_fraction * 100.0,
            assignment.estimated_completion_ms
        );
    }

    // Run comprehensive benchmarks
    let benchmark_results = BenchmarkRunner::run_quick_validation().await?;
    println!("Benchmark completed: {} tests, average scaling efficiency: {:.2}%",
        benchmark_results.performance_summary.total_tests,
        benchmark_results.performance_summary.average_scaling_efficiency * 100.0
    );

    Ok(())
}

Rust: Geometric Algebra

use amari_core::{Multivector, basis::Basis, rotor::Rotor};

// 3D Euclidean Clifford algebra
type Cl3 = Multivector<3, 0, 0>;

// Create basis vectors
let e1: Cl3 = Basis::e1();
let e2: Cl3 = Basis::e2();

// Geometric product: e1 * e2 = e1 ∧ e2 (bivector)
let e12 = e1.geometric_product(&e2);

// Create rotor for 90° rotation in xy-plane
let rotor = Rotor::from_bivector(&e12, std::f64::consts::PI / 2.0);

// Apply rotation: e1 → e2
let rotated = rotor.apply(&e1);

Rust: Tropical-Dual-Clifford System

use amari_fusion::TropicalDualClifford;
use amari_dual::DualNumber;
use amari_tropical::TropicalNumber;

// Create from logits (common in ML applications)
let logits = vec![1.5, 2.0, 0.8, 1.2];
let tdc = TropicalDualClifford::<f64, 4>::from_logits(&logits);

// Evaluate using all three algebras simultaneously
let other = TropicalDualClifford::from_logits(&[2.0, 1.5, 1.0, 0.9]);
let evaluation = tdc.evaluate(&other);

// Extract features from each algebra
let tropical_features = tdc.extract_tropical_features(); // Fast path-finding
let dual_features = tdc.extract_dual_features();         // Automatic gradients
let clifford_geom = tdc.clifford;                        // Geometric relationships

// Perform sensitivity analysis
let sensitivity = tdc.sensitivity_analysis();
let most_sensitive = sensitivity.most_sensitive(2);

println!("Combined score: {}", evaluation.combined_score);
println!("Most sensitive components: {:?}", most_sensitive);

JavaScript/TypeScript: Mathematical Computing

import init, { WasmMultivector, WasmTropicalNumber, WasmDualNumber } from '@justinelliottcobb/amari-wasm';

async function main() {
  // Initialize the WASM module
  await init();

  // Geometric Algebra: Create and rotate vectors
  const e1 = WasmMultivector.basisVector(0);
  const e2 = WasmMultivector.basisVector(1);
  const bivector = e1.geometricProduct(e2);
  console.log('Geometric product:', bivector.toString());

  // Tropical Algebra: Neural network operations
  const trop1 = new WasmTropicalNumber(3.0);
  const trop2 = new WasmTropicalNumber(5.0);
  const sum = trop1.tropicalAdd(trop2); // max(3, 5) = 5
  const product = trop1.tropicalMul(trop2); // 3 + 5 = 8
  console.log('Tropical operations:', sum.getValue(), product.getValue());

  // Automatic Differentiation: Compute derivatives
  const x = new WasmDualNumber(2.0, 1.0);
  const xSquared = x.mul(x); // f(x) = x², f'(x) = 2x
  console.log('f(2) =', xSquared.getReal(), "f'(2) =", xSquared.getDual());

  // Clean up WASM memory
  e1.free(); e2.free(); bivector.free();
  trop1.free(); trop2.free(); sum.free(); product.free();
  x.free(); xSquared.free();
}

main();

Architecture

Crates

amari-core: Core Clifford algebra types and CPU implementations
amari-tropical: Tropical (max-plus) algebra for neural networks
amari-dual: Dual numbers for automatic differentiation
amari-fusion: Unified Tropical-Dual-Clifford system
amari-info-geom: Information geometry and statistical manifolds
amari-wasm: WASM bindings for TypeScript/JavaScript
amari-gpu: Multi-GPU acceleration with intelligent load balancing
amari-automata: Cellular automata with geometric algebra
amari-network: Graph neural networks and network analysis
amari-relativistic: Spacetime algebra and relativistic physics
amari-enumerative: Enumerative geometry and algebraic curves
amari-optimization: Gradient-based optimization algorithms

Key Types

// Multivector in Clifford algebra Cl(P,Q,R)
Multivector<const P: usize, const Q: usize, const R: usize>

// Tropical-Dual-Clifford unified system
TropicalDualClifford<T: Float, const DIM: usize>

// Dual numbers for automatic differentiation
DualNumber<T: Float>

// Tropical numbers for max-plus algebra
TropicalNumber<T: Float>

// Common algebras
type Cl3 = Multivector<3, 0, 0>;  // 3D Euclidean
type Spacetime = Multivector<1, 3, 0>;  // Minkowski spacetime

Multi-Backend Precision Architecture

Automatic Backend Selection

Amari provides intelligent backend selection for high-precision arithmetic:

// Same API, different backends automatically selected:

// For native builds (optimal performance)
cargo build --features native-precision  // Uses rug (GMP/MPFR)

// For WASM builds (maximum compatibility)
cargo build --target wasm32-unknown-unknown --features wasm-precision  // Uses dashu

// Auto-selection (recommended)
cargo build --features high-precision  // Chooses best backend for target

Backend Characteristics

Backend	Platform	Performance	Dependencies	Use Case
rug	Native	Ultimate	GMP/MPFR (C libraries)	High-performance computing, research
dashu	WASM	Excellent	Pure Rust	Web apps, edge computing, universality

Mathematical Consistency

Both backends provide:

Identical API: Same function signatures across platforms
Numerical Accuracy: Configurable precision with orbital-grade tolerance
Mathematical Correctness: All relativistic calculations preserve physical invariants
Feature Parity: Full support for spacecraft orbital mechanics in both environments

WebAssembly Deployment Example

// Compile for WASM with high-precision arithmetic
#[cfg(target_arch = "wasm32")]
use amari_relativistic::precision::StandardFloat; // Uses dashu backend

// Native compilation automatically uses rug
#[cfg(not(target_arch = "wasm32"))]
use amari_relativistic::precision::StandardFloat; // Uses rug backend

// Same code works everywhere!
let spacecraft_trajectory = propagate_orbital_mechanics(
    initial_conditions,
    StandardFloat::orbital_tolerance(), // 1e-12 precision
)?;

Mathematical Foundation

Tropical-Dual-Clifford System

The fusion of three algebraic systems:

Tropical Algebra (Max-Plus)

a ⊕ b = max(a, b)    // Tropical addition
a ⊙ b = a + b        // Tropical multiplication

Applications: Path optimization, sequence decoding, dynamic programming
Benefits: Converts exponential operations to linear max operations

Dual Numbers

a + εb where ε² = 0
(a + εb) + (c + εd) = (a + c) + ε(b + d)
(a + εb) × (c + εd) = ac + ε(ad + bc)

Applications: Automatic differentiation, gradient computation
Benefits: Exact derivatives without finite differences or computational graphs

Clifford Algebra

ab = a·b + a∧b      // Geometric product

Applications: Rotations, reflections, geometric transformations
Benefits: Unified treatment of scalars, vectors, bivectors, trivectors

Unified TDC Operations

The fusion system enables simultaneous computation across all three algebras:

Tropical Phase: Fast approximation using max-plus operations
Dual Phase: Exact computation with automatic gradients
Clifford Phase: Geometric refinement and spatial reasoning

Clifford Algebra Cl(P,Q,R)

P: Basis vectors with e²ᵢ = +1
Q: Basis vectors with e²ᵢ = -1
R: Basis vectors with e²ᵢ = 0

Information Geometry

Fisher Information Metric: Riemannian metric on statistical manifolds
α-Connections: Generalized connections parameterized by α ∈ [-1,1]
Dually Flat Manifolds: Manifolds with e-connection (α=+1) and m-connection (α=-1)
Bregman Divergences: Information-geometric divergences
Amari-Chentsov Tensor: Fundamental tensor structure

Use Cases

Scientific Computing

Computer Graphics: 3D rotations and transformations using rotors
Physics Simulations: Geometric algebra for electromagnetic fields
Relativistic Physics: Spacetime calculations and orbital mechanics
Astrophysics: Schwarzschild metric and geodesic integration

Machine Learning

Neural Networks: Tropical algebra for efficient network operations
Automatic Differentiation: Forward-mode AD with dual numbers
Statistical Manifolds: Information geometry and natural gradients
Geometric Deep Learning: Operations on non-Euclidean data

Optimization

Path Optimization: Tropical algebra for shortest path problems
Gradient-Based Optimization: Hybrid tropical-dual-Clifford optimization
Sequence Analysis: Efficient decoding using tropical Viterbi
Mathematical Optimization: Multi-algebraic system integration

Research Applications

Information Geometry: Statistical manifold computations
Quantum Computing: Clifford group operations
Crystallography: Symmetry group calculations
Tropical Geometry: Max-plus linear algebra and optimization
Computational Algebra: Multi-algebraic system integration
Neural Architecture Search: Gradient-based optimization with geometric constraints

Examples

Tropical Sequence Decoding

use amari_tropical::viterbi::ViterbiDecoder;

// Efficient Viterbi algorithm using tropical algebra
let transitions = create_transition_matrix();
let emissions = create_emission_matrix();
let observations = vec![0, 1, 2, 1, 0];

let decoder = ViterbiDecoder::new(&transitions, &emissions);
let best_path = decoder.decode(&observations);

Automatic Differentiation

use amari_dual::{DualNumber, functions::softmax};

// Forward-mode autodiff
let inputs: Vec<DualNumber<f64>> = vec![
    DualNumber::variable(1.0),
    DualNumber::variable(2.0),
    DualNumber::variable(0.5),
];

let output = softmax(&inputs);
// output[i].real contains the value
// output[i].dual contains the gradient

Information Geometry

use amari_info_geom::{bregman_divergence, kl_divergence};

// Bregman divergence with quadratic potential
let phi = |mv: &Multivector<3,0,0>| mv.norm_squared();
let divergence = bregman_divergence(phi, &p, &q)?;

Optimization

use amari_fusion::{TropicalDualClifford, optimizer::TDCOptimizer};

// Optimization using all three algebras
let initial_params = vec![0.1, 0.5, -0.2, 0.8];
let tdc = TropicalDualClifford::<f64, 4>::from_logits(&initial_params);

let optimizer = TDCOptimizer::new()
    .with_tropical_warmup(5)     // Fast tropical approximation
    .with_dual_refinement(10)    // Exact dual gradients
    .with_clifford_projection(); // Geometric constraints

let result = optimizer.optimize(&tdc, &target_function)?;

Building

Prerequisites

Rust 1.75+ with cargo
Node.js 16+ (for TypeScript bindings)
wasm-pack (installed automatically by build script)

Build All Tests

cargo test --workspace

Documentation

cargo doc --workspace --open

API documentation will be available at target/doc/amari_core/index.html

Performance

The library is optimized for high-performance applications:

SIMD: Vectorized operations where supported
Cache Alignment: 64-byte aligned data structures
Const Generics: Zero-cost abstractions for dimensions
GPU Fallback: Automatic CPU/GPU dispatch based on workload size
Batch Operations: Efficient batch processing for large datasets

Integration Status

Amari provides three deployment targets with varying levels of integration across crates:

Target	Description	Coverage
Rust	Native library	All crates fully integrated
WASM	Web browsers/Node.js	Complete coverage for all domain crates
GPU	Hardware acceleration	Core + network + info-geom + relativistic modules

For detailed architecture documentation, see docs/architecture/v0.9.6-multi-gpu-design.md.

Documentation

Comprehensive documentation is available in the docs/ directory:

Release Process: Current release procedures
Multi-GPU Design: Multi-GPU architecture details
Publishing Guide: Publishing to npm and crates.io
Technical Documentation: Mathematical rigor, phantom types, formal verification

See docs/README.md for the complete documentation index.

Contributing

Contributions are welcome. For development setup:

git clone https://github.com/justinelliottcobb/Amari.git
cd amari
cargo test --workspace

License

This project is licensed under either of:

Apache License, Version 2.0 (LICENSE-APACHE)
MIT License (LICENSE-MIT)

at your option.

Acknowledgments

Inspired by the geometric algebra community and research in Information Geometry
Built with modern Rust performance idioms and WebAssembly best practices
Named after Shun-ichi Amari's contributions to Information Geometry

Support

Issues: GitHub Issues
Discussions: GitHub Discussions
Documentation: API Docs

"Geometry is the art of correct reasoning from incorrectly drawn figures." - Henri Poincaré

amari 0.9.9

Amari v0.9.8

Features

Core Mathematical Systems

Multi-GPU Infrastructure

High-Precision Arithmetic

Platform Support

Installation

Rust Crates

JavaScript/TypeScript (WebAssembly)

Quick Start

Rust: Spacecraft Orbital Mechanics

Rust: Multi-GPU Load Balancing

Rust: Geometric Algebra

Rust: Tropical-Dual-Clifford System

JavaScript/TypeScript: Mathematical Computing

Architecture

Crates

Key Types

Multi-Backend Precision Architecture

Automatic Backend Selection

Backend Characteristics

Mathematical Consistency

WebAssembly Deployment Example

Mathematical Foundation

Tropical-Dual-Clifford System

Tropical Algebra (Max-Plus)

Dual Numbers

Clifford Algebra

Unified TDC Operations

Clifford Algebra Cl(P,Q,R)

Information Geometry

Use Cases

Scientific Computing

Machine Learning

Optimization

Research Applications

Examples

Tropical Sequence Decoding

Automatic Differentiation

Information Geometry

Optimization

Building

Prerequisites

Build All Tests

Documentation

Performance

Integration Status

Documentation

Contributing

License

Acknowledgments

Support