SparseRationalLearning

A Julia package for learning rational functions with sparsity from data.

Introduction

SparseRationalLearning provides algorithms for learning rational functions of the form P(x)/Q(x) from data, where P and Q are polynomials. By imposing sparsity-promoting regularization, the package learns models that are both compact and interpretable.

Installation

# Navigate to the project directory and start Julia REPL
julia

# Activate the project environment
using Pkg
Pkg.activate(".")

# Install dependencies
Pkg.instantiate()

Quick Start

using SparseRationalLearning

# Generate sample data
X = rand(100, 2) .* 5.0 # 100 samples, 2 features
y = (1.5 .* X[:,1] .* X[:,2] .+ 0.8 .* X[:,1].^2) ./ (1.0 .+ 0.2 .* X[:,2].^2)

#  Train a sparse rational model using the L1-based method (fast)
a, b, labels = train_l1(X, y, 2, 1.0)

# Display the learned symbolic expression
println(sparse_rational_expression(a, b, labels, threshold=0.01))

# Compute the coefficient of determination (R²)
r2, _ = calculate_r2(X, y, a, b, 2)
println("R² = $r2")

API Reference

Training Functions

train_l0(X, y, degree, lambda; M=50.0, time_limit=60.0)

Fits a sparse rational function model using L0 regularization via mixed-integer programming.

Parameters:

• X: Feature matrix of size (n_samples, n_features)
• y: Target vector
• degree: Maximum polynomial degree
• lambda: Sparsity regularization parameter
• M: Big-M constant used in the mixed-integer formulation
• time_limit: Solver time limit in seconds

Returns: (a, b, labels) — numerator coefficients, denominator coefficients, and basis-function labels

---

train_l1(X, y, degree, lambda)

Fits a sparse rational function model using L1 regularization (Lasso).

Parameters:

• X: Feature matrix
• y: Target vector
• degree: Maximum polynomial degree
• lambda: L1 regularization coefficient

Returns: (a, b, labels) — numerator coefficients, denominator coefficients, and basis-function labels

---

Utility Functions

Function	Description
sparse_rational_expression(a, b, labels; threshold)	Generate rational function expression string
calculate_r2(X, y_true, a, b, degree)	Calculate the coefficient of determination (R²)
coefficients_threshold(a, b; threshold)	Truncate small coefficients to 0

Algorithm Comparison

Feature	L0 Method	L1 Method
Solver	HiGHS	SCS
Sparsity	More precise	Approximate
Speed	Slower	Faster
Recommended use	Small-scale problems	Large-scale problems

Running Tests

using Pkg
Pkg.test()

Dependencies

• Julia >= 1.12
• Convex.jl - Convex optimization modeling
• HiGHS.jl - Mixed-integer programming (MIP) solver
• SCS.jl - Conic optimization solver
• Combinatorics.jl - Combinatorial utilities

License

MIT License

Citation

If you find this code or our methodology useful in your research, please consider citing our paper:

@article{li2026SpareRational,
  title={Sparse Rational Learning for Interpretable Aerodynamic Surrogate Modeling},
  author={Li, Jing and Luo, Changtong},
  journal={TBD},
  year={2026}
}

---

Co-author: @lijing-creator

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