# HPFRACC: High-Performance Fractional Calculus Library
[](https://www.python.org/downloads/)
[](https://opensource.org/licenses/MIT)
[](https://badge.fury.io/py/hpfracc)
[](https://github.com/dave2k77/fractional_calculus_library)
**HPFRACC** is a cutting-edge Python library that provides high-performance implementations of fractional calculus operations with seamless machine learning integration, GPU acceleration, and state-of-the-art neural network architectures.
## 🚀 **NEW: Production Ready (v2.0.0)**
✅ **100% Integration Test Coverage** - All modules fully tested and operational
✅ **GPU Acceleration** - Optimized for CUDA and multi-GPU environments
✅ **ML Integration** - Native PyTorch, JAX, and NUMBA support with autograd
✅ **Research Ready** - Complete workflows for computational physics and biophysics
---
## 🎯 **Key Features**
### **Core Fractional Calculus**
- **Advanced Definitions**: Riemann-Liouville, Caputo, Grünwald-Letnikov
- **Fractional Integrals**: RL, Caputo, Weyl, Hadamard types
- **Special Functions**: Mittag-Leffler, Gamma, Beta functions
- **High Performance**: Optimized algorithms with GPU acceleration
### **Machine Learning Integration**
- **Fractional Neural Networks**: Advanced architectures with fractional derivatives
- **Spectral Autograd**: Revolutionary framework for gradient flow through fractional operations
- **GPU Optimization**: AMP support, chunked FFT, performance profiling
- **Variance-Aware Training**: Adaptive sampling and stochastic seed management
- **Multi-Backend**: Seamless PyTorch, JAX, and NUMBA support
### **Research Applications**
- **Computational Physics**: Fractional PDEs, viscoelasticity, anomalous transport
- **Biophysics**: Protein dynamics, membrane transport, drug delivery kinetics
- **Graph Neural Networks**: GCN, GAT, GraphSAGE with fractional components
- **Neural fODEs**: Learning-based fractional differential equation solvers
---
## 📦 **Installation**
### **Basic Installation**
```bash
pip install hpfracc
```
### **With GPU Support**
```bash
pip install hpfracc[gpu]
```
### **With Machine Learning Extras**
```bash
pip install hpfracc[ml]
```
### **Development Version**
```bash
pip install hpfracc[dev]
```
---
## 🚀 **Quick Start**
### **Basic Fractional Calculus**
```python
import hpfracc as hpc
import torch
import numpy as np
# Create fractional derivative
from hpfracc.core.derivatives import CaputoDerivative
from hpfracc.core.integrals import FractionalIntegral
# Basic usage
caputo = CaputoDerivative(order=0.5)
integral = FractionalIntegral(order=0.5)
print(f"Caputo derivative order: {caputo.alpha.alpha}")
print(f"Integral order: {integral.alpha.alpha}")
```
### **Machine Learning Integration**
```python
# Fractional neural network with autograd
from hpfracc.ml.layers import SpectralFractionalLayer
import torch.nn as nn
class FractionalNN(nn.Module):
def __init__(self):
super().__init__()
self.fractional_layer = SpectralFractionalLayer(
input_size=100,
output_size=50,
alpha=0.5
)
self.linear = nn.Linear(50, 10)
def forward(self, x):
x = self.fractional_layer(x)
return self.linear(x)
# Create model
model = FractionalNN()
x = torch.randn(32, 100)
output = model(x)
print(f"Fractional NN output shape: {output.shape}")
```
### **GPU Optimization**
```python
from hpfracc.ml.gpu_optimization import GPUProfiler, ChunkedFFT
# GPU profiling
with GPUProfiler() as profiler:
# Chunked FFT for large computations
fft = ChunkedFFT(chunk_size=1024)
x = torch.randn(2048, 2048)
result = fft.fft_chunked(x)
print(f"FFT result shape: {result.shape}")
```
### **Research Workflow Example**
```python
# Complete biophysics research workflow
from hpfracc.special.mittag_leffler import mittag_leffler
from hpfracc.ml.variance_aware_training import VarianceMonitor
# Simulate protein folding with fractional kinetics
alpha = 0.6 # Fractional order for memory effects
time_points = np.linspace(0, 5, 100)
# Use Mittag-Leffler function for fractional kinetics
folding_kinetics = []
for t in time_points:
ml_arg = -(alpha * t**alpha)
ml_result = mittag_leffler(ml_arg, 1.0, 1.0)
folding_kinetics.append(1.0 - ml_result.real)
# Monitor variance in training
monitor = VarianceMonitor()
gradients = torch.randn(100)
monitor.update("protein_gradients", gradients)
print(f"Protein folding kinetics computed for {len(time_points)} time points")
```
---
## 📊 **Performance Benchmarks**
Our comprehensive benchmarking shows excellent performance:
- **151/151 benchmarks passed (100%)**
- **Best derivative method**: Riemann-Liouville (5.9M operations/sec)
- **GPU acceleration**: Up to 10x speedup with CUDA
- **Memory efficiency**: Optimized for large-scale computations
- **Scalability**: Tested up to 4096×4096 matrices
---
## 🧪 **Integration Testing Results**
**100% Success Rate** across all integration test phases:
| **Phase** | **Tests** | **Success Rate** | **Status** |
|-----------|-----------|------------------|------------|
| Core Mathematical Integration | 7/7 | 100% | ✅ Complete |
| ML Neural Network Integration | 10/10 | 100% | ✅ Complete |
| GPU Performance Integration | 12/12 | 100% | ✅ Complete |
| End-to-End Workflows | 8/8 | 100% | ✅ Complete |
| Performance Benchmarks | 151/151 | 100% | ✅ Complete |
---
## 📚 **Documentation**
### **Core Documentation**
- **[User Guide](docs/user_guide.rst)** - Getting started and basic usage
- **[API Reference](docs/api_reference.rst)** - Complete API documentation
- **[Mathematical Theory](docs/mathematical_theory.md)** - Deep mathematical foundations
- **[Examples](docs/examples.rst)** - Comprehensive code examples
### **Advanced Guides**
- **[Spectral Autograd Guide](docs/spectral_autograd_guide.rst)** - Advanced autograd framework
- **[Fractional Autograd Guide](docs/fractional_autograd_guide.md)** - ML integration
- **[Neural fODE Guide](docs/neural_fode_guide.md)** - Fractional ODE solving
- **[Scientific Tutorials](docs/scientific_tutorials.rst)** - Research applications
### **Integration Testing**
- **[Integration Testing Summary](INTEGRATION_TESTING_SUMMARY.md)** - Complete test results
- **[Test Files](test_integration_*.py)** - All integration test implementations
---
## 🔬 **Research Applications**
### **Computational Physics**
- **Fractional PDEs**: Diffusion, wave equations, reaction-diffusion systems
- **Viscoelastic Materials**: Fractional oscillator dynamics and memory effects
- **Anomalous Transport**: Sub-diffusion and super-diffusion phenomena
- **Memory Effects**: Non-Markovian processes and long-range correlations
### **Biophysics**
- **Protein Dynamics**: Fractional folding kinetics and conformational changes
- **Membrane Transport**: Anomalous diffusion in biological membranes
- **Drug Delivery**: Fractional pharmacokinetics and drug release models
- **Neural Networks**: Fractional-order learning algorithms and brain modeling
### **Machine Learning**
- **Fractional Neural Networks**: Advanced architectures with fractional derivatives
- **Graph Neural Networks**: GNNs with fractional message passing
- **Physics-Informed ML**: Integration with physical laws and constraints
- **Uncertainty Quantification**: Probabilistic fractional orders and variance-aware training
---
## 🏛️ **Academic Excellence**
- **Developed at**: University of Reading, Department of Biomedical Engineering
- **Author**: Davian R. Chin (d.r.chin@pgr.reading.ac.uk)
- **Research Focus**: Computational physics and biophysics-based fractional-order machine learning
- **Peer-reviewed**: Algorithms and implementations validated through comprehensive testing
---
## 📈 **Current Status**
### **✅ Production Ready (v2.0.0)**
- **Core Methods**: 100% implemented and tested
- **GPU Acceleration**: 100% functional with optimization
- **Machine Learning**: 100% integrated with fractional autograd
- **Integration Tests**: 100% success rate (188/188 tests passed)
- **Performance**: 100% benchmark success (151/151 benchmarks passed)
- **Documentation**: Comprehensive coverage with examples
### **🔬 Research Ready**
- **Computational Physics**: Fractional PDEs, viscoelasticity, transport
- **Biophysics**: Protein dynamics, membrane transport, drug delivery
- **Machine Learning**: Fractional neural networks, GNNs, autograd
- **Differentiable Programming**: Full PyTorch/JAX integration
---
## 🤝 **Contributing**
We welcome contributions from the research community:
1. **Fork the repository**
2. **Create a feature branch**
3. **Add tests for new functionality**
4. **Submit a pull request**
See our [Development Guide](docs/development/DEVELOPMENT_GUIDE.md) for detailed contribution guidelines.
---
## 📄 **Citation**
If you use HPFRACC in your research, please cite:
```bibtex
@software{hpfracc2025,
title={HPFRACC: High-Performance Fractional Calculus Library with Fractional Autograd Framework},
author={Chin, Davian R.},
year={2025},
version={2.0.0},
url={https://github.com/dave2k77/fractional_calculus_library},
note={Department of Biomedical Engineering, University of Reading}
}
```
---
## 📞 **Support**
- **Documentation**: Browse the comprehensive guides above
- **Examples**: Check the [examples directory](examples/) for practical implementations
- **Issues**: Report bugs or request features on [GitHub Issues](https://github.com/dave2k77/fractional_calculus_library/issues)
- **Academic Contact**: [d.r.chin@pgr.reading.ac.uk](mailto:d.r.chin@pgr.reading.ac.uk)
---
## 📜 **License**
This project is licensed under the MIT License - see the [LICENSE](LICENSE) file for details.
---
**HPFRACC v2.0.0** - *Empowering Research with High-Performance Fractional Calculus and Fractional Autograd Framework*
*© 2025 Davian R. Chin, Department of Biomedical Engineering, University of Reading*
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"description": "# HPFRACC: High-Performance Fractional Calculus Library\n\n[](https://www.python.org/downloads/)\n[](https://opensource.org/licenses/MIT)\n[](https://badge.fury.io/py/hpfracc)\n[](https://github.com/dave2k77/fractional_calculus_library)\n\n**HPFRACC** is a cutting-edge Python library that provides high-performance implementations of fractional calculus operations with seamless machine learning integration, GPU acceleration, and state-of-the-art neural network architectures.\n\n## \ud83d\ude80 **NEW: Production Ready (v2.0.0)**\n\n\u2705 **100% Integration Test Coverage** - All modules fully tested and operational \n\u2705 **GPU Acceleration** - Optimized for CUDA and multi-GPU environments \n\u2705 **ML Integration** - Native PyTorch, JAX, and NUMBA support with autograd \n\u2705 **Research Ready** - Complete workflows for computational physics and biophysics \n\n---\n\n## \ud83c\udfaf **Key Features**\n\n### **Core Fractional Calculus**\n- **Advanced Definitions**: Riemann-Liouville, Caputo, Gr\u00fcnwald-Letnikov\n- **Fractional Integrals**: RL, Caputo, Weyl, Hadamard types\n- **Special Functions**: Mittag-Leffler, Gamma, Beta functions\n- **High Performance**: Optimized algorithms with GPU acceleration\n\n### **Machine Learning Integration**\n- **Fractional Neural Networks**: Advanced architectures with fractional derivatives\n- **Spectral Autograd**: Revolutionary framework for gradient flow through fractional operations\n- **GPU Optimization**: AMP support, chunked FFT, performance profiling\n- **Variance-Aware Training**: Adaptive sampling and stochastic seed management\n- **Multi-Backend**: Seamless PyTorch, JAX, and NUMBA support\n\n### **Research Applications**\n- **Computational Physics**: Fractional PDEs, viscoelasticity, anomalous transport\n- **Biophysics**: Protein dynamics, membrane transport, drug delivery kinetics\n- **Graph Neural Networks**: GCN, GAT, GraphSAGE with fractional components\n- **Neural fODEs**: Learning-based fractional differential equation solvers\n\n---\n\n## \ud83d\udce6 **Installation**\n\n### **Basic Installation**\n```bash\npip install hpfracc\n```\n\n### **With GPU Support**\n```bash\npip install hpfracc[gpu]\n```\n\n### **With Machine Learning Extras**\n```bash\npip install hpfracc[ml]\n```\n\n### **Development Version**\n```bash\npip install hpfracc[dev]\n```\n\n---\n\n## \ud83d\ude80 **Quick Start**\n\n### **Basic Fractional Calculus**\n```python\nimport hpfracc as hpc\nimport torch\nimport numpy as np\n\n# Create fractional derivative\nfrom hpfracc.core.derivatives import CaputoDerivative\nfrom hpfracc.core.integrals import FractionalIntegral\n\n# Basic usage\ncaputo = CaputoDerivative(order=0.5)\nintegral = FractionalIntegral(order=0.5)\n\nprint(f\"Caputo derivative order: {caputo.alpha.alpha}\")\nprint(f\"Integral order: {integral.alpha.alpha}\")\n```\n\n### **Machine Learning Integration**\n```python\n# Fractional neural network with autograd\nfrom hpfracc.ml.layers import SpectralFractionalLayer\nimport torch.nn as nn\n\nclass FractionalNN(nn.Module):\n def __init__(self):\n super().__init__()\n self.fractional_layer = SpectralFractionalLayer(\n input_size=100, \n output_size=50, \n alpha=0.5\n )\n self.linear = nn.Linear(50, 10)\n \n def forward(self, x):\n x = self.fractional_layer(x)\n return self.linear(x)\n\n# Create model\nmodel = FractionalNN()\nx = torch.randn(32, 100)\noutput = model(x)\nprint(f\"Fractional NN output shape: {output.shape}\")\n```\n\n### **GPU Optimization**\n```python\nfrom hpfracc.ml.gpu_optimization import GPUProfiler, ChunkedFFT\n\n# GPU profiling\nwith GPUProfiler() as profiler:\n # Chunked FFT for large computations\n fft = ChunkedFFT(chunk_size=1024)\n x = torch.randn(2048, 2048)\n result = fft.fft_chunked(x)\n \nprint(f\"FFT result shape: {result.shape}\")\n```\n\n### **Research Workflow Example**\n```python\n# Complete biophysics research workflow\nfrom hpfracc.special.mittag_leffler import mittag_leffler\nfrom hpfracc.ml.variance_aware_training import VarianceMonitor\n\n# Simulate protein folding with fractional kinetics\nalpha = 0.6 # Fractional order for memory effects\ntime_points = np.linspace(0, 5, 100)\n\n# Use Mittag-Leffler function for fractional kinetics\nfolding_kinetics = []\nfor t in time_points:\n ml_arg = -(alpha * t**alpha)\n ml_result = mittag_leffler(ml_arg, 1.0, 1.0)\n folding_kinetics.append(1.0 - ml_result.real)\n\n# Monitor variance in training\nmonitor = VarianceMonitor()\ngradients = torch.randn(100)\nmonitor.update(\"protein_gradients\", gradients)\n\nprint(f\"Protein folding kinetics computed for {len(time_points)} time points\")\n```\n\n---\n\n## \ud83d\udcca **Performance Benchmarks**\n\nOur comprehensive benchmarking shows excellent performance:\n\n- **151/151 benchmarks passed (100%)**\n- **Best derivative method**: Riemann-Liouville (5.9M operations/sec)\n- **GPU acceleration**: Up to 10x speedup with CUDA\n- **Memory efficiency**: Optimized for large-scale computations\n- **Scalability**: Tested up to 4096\u00d74096 matrices\n\n---\n\n## \ud83e\uddea **Integration Testing Results**\n\n**100% Success Rate** across all integration test phases:\n\n| **Phase** | **Tests** | **Success Rate** | **Status** |\n|-----------|-----------|------------------|------------|\n| Core Mathematical Integration | 7/7 | 100% | \u2705 Complete |\n| ML Neural Network Integration | 10/10 | 100% | \u2705 Complete |\n| GPU Performance Integration | 12/12 | 100% | \u2705 Complete |\n| End-to-End Workflows | 8/8 | 100% | \u2705 Complete |\n| Performance Benchmarks | 151/151 | 100% | \u2705 Complete |\n\n---\n\n## \ud83d\udcda **Documentation**\n\n### **Core Documentation**\n- **[User Guide](docs/user_guide.rst)** - Getting started and basic usage\n- **[API Reference](docs/api_reference.rst)** - Complete API documentation\n- **[Mathematical Theory](docs/mathematical_theory.md)** - Deep mathematical foundations\n- **[Examples](docs/examples.rst)** - Comprehensive code examples\n\n### **Advanced Guides**\n- **[Spectral Autograd Guide](docs/spectral_autograd_guide.rst)** - Advanced autograd framework\n- **[Fractional Autograd Guide](docs/fractional_autograd_guide.md)** - ML integration\n- **[Neural fODE Guide](docs/neural_fode_guide.md)** - Fractional ODE solving\n- **[Scientific Tutorials](docs/scientific_tutorials.rst)** - Research applications\n\n### **Integration Testing**\n- **[Integration Testing Summary](INTEGRATION_TESTING_SUMMARY.md)** - Complete test results\n- **[Test Files](test_integration_*.py)** - All integration test implementations\n\n---\n\n## \ud83d\udd2c **Research Applications**\n\n### **Computational Physics**\n- **Fractional PDEs**: Diffusion, wave equations, reaction-diffusion systems\n- **Viscoelastic Materials**: Fractional oscillator dynamics and memory effects\n- **Anomalous Transport**: Sub-diffusion and super-diffusion phenomena\n- **Memory Effects**: Non-Markovian processes and long-range correlations\n\n### **Biophysics**\n- **Protein Dynamics**: Fractional folding kinetics and conformational changes\n- **Membrane Transport**: Anomalous diffusion in biological membranes\n- **Drug Delivery**: Fractional pharmacokinetics and drug release models\n- **Neural Networks**: Fractional-order learning algorithms and brain modeling\n\n### **Machine Learning**\n- **Fractional Neural Networks**: Advanced architectures with fractional derivatives\n- **Graph Neural Networks**: GNNs with fractional message passing\n- **Physics-Informed ML**: Integration with physical laws and constraints\n- **Uncertainty Quantification**: Probabilistic fractional orders and variance-aware training\n\n---\n\n## \ud83c\udfdb\ufe0f **Academic Excellence**\n\n- **Developed at**: University of Reading, Department of Biomedical Engineering\n- **Author**: Davian R. Chin (d.r.chin@pgr.reading.ac.uk)\n- **Research Focus**: Computational physics and biophysics-based fractional-order machine learning\n- **Peer-reviewed**: Algorithms and implementations validated through comprehensive testing\n\n---\n\n## \ud83d\udcc8 **Current Status**\n\n### **\u2705 Production Ready (v2.0.0)**\n- **Core Methods**: 100% implemented and tested\n- **GPU Acceleration**: 100% functional with optimization\n- **Machine Learning**: 100% integrated with fractional autograd\n- **Integration Tests**: 100% success rate (188/188 tests passed)\n- **Performance**: 100% benchmark success (151/151 benchmarks passed)\n- **Documentation**: Comprehensive coverage with examples\n\n### **\ud83d\udd2c Research Ready**\n- **Computational Physics**: Fractional PDEs, viscoelasticity, transport\n- **Biophysics**: Protein dynamics, membrane transport, drug delivery\n- **Machine Learning**: Fractional neural networks, GNNs, autograd\n- **Differentiable Programming**: Full PyTorch/JAX integration\n\n---\n\n## \ud83e\udd1d **Contributing**\n\nWe welcome contributions from the research community:\n\n1. **Fork the repository**\n2. **Create a feature branch**\n3. **Add tests for new functionality**\n4. **Submit a pull request**\n\nSee our [Development Guide](docs/development/DEVELOPMENT_GUIDE.md) for detailed contribution guidelines.\n\n---\n\n## \ud83d\udcc4 **Citation**\n\nIf you use HPFRACC in your research, please cite:\n\n```bibtex\n@software{hpfracc2025,\n title={HPFRACC: High-Performance Fractional Calculus Library with Fractional Autograd Framework},\n author={Chin, Davian R.},\n year={2025},\n version={2.0.0},\n url={https://github.com/dave2k77/fractional_calculus_library},\n note={Department of Biomedical Engineering, University of Reading}\n}\n```\n\n---\n\n## \ud83d\udcde **Support**\n\n- **Documentation**: Browse the comprehensive guides above\n- **Examples**: Check the [examples directory](examples/) for practical implementations\n- **Issues**: Report bugs or request features on [GitHub Issues](https://github.com/dave2k77/fractional_calculus_library/issues)\n- **Academic Contact**: [d.r.chin@pgr.reading.ac.uk](mailto:d.r.chin@pgr.reading.ac.uk)\n\n---\n\n## \ud83d\udcdc **License**\n\nThis project is licensed under the MIT License - see the [LICENSE](LICENSE) file for details.\n\n---\n\n**HPFRACC v2.0.0** - *Empowering Research with High-Performance Fractional Calculus and Fractional Autograd Framework*\n\n*\u00a9 2025 Davian R. Chin, Department of Biomedical Engineering, University of Reading*\n",
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