VariationalInverse.jl

VariationalInverse.jl is a lightweight Julia package designed to solve inverse mapping problems using variational optimization. It transforms the task of finding an inverse function into an unconstrained optimization problem, powered by the robust solvers in Optim.jl.

💡 Core Principle

For a given mapping $\Psi$, the inverse $x = \Psi^{-1}(y)$ is found by minimizing the following objective function: $$f(x) = \Psi(x) - \langle x, y \rangle$$

By default, the package uses the L-BFGS algorithm, which is highly efficient for high-dimensional problems.

✨ Features

• Multi-Dimensional Support: Works seamlessly with scalars, vectors, and matrices (processed column-wise).
• Customizable Optimizers: Full compatibility with Optim.jl solvers (e.g., LBFGS, GradientDescent, Newton).
• Solver Interface: Use the InverseSolver struct to pre-configure optimization settings for repeated use.

📦 Installation

This package is currently in development. You can install it via the Julia REPL:

using Pkg
Pkg.add(url="[https://github.com/VaneBlien/VariationalInverse.jl](https://github.com/VaneBlien/VariationalInverse.jl)")

🚀 Quick Start

1. Basic Usage (Scalar)

Find the inverse of $f(x) = x^2$ at $y = 4.0$:

using VariationalInverse

Ψ(x) = x^2 
y = 4.0
x_inv = inverse(Ψ, y)
println("Result: ", x_inv) # Expected: ~2.0

2. Vector and Matrix Inputs

using LinearAlgebra

# Vector input
Ψ_vec(x) = 0.5 * dot(x, x)
y_vec = [1.0, 2.0, 3.0]
x_vec = inverse(Ψ_vec, y_vec)

# Matrix input (computes inverse for each column)
Y = [1.0 2.0; 3.0 4.0]
X = inverse(Ψ_vec, Y)

3. Using the Solver Pattern

For production workflows, pre-define your solver settings:

solver = InverseSolver(Ψ_vec, method=LBFGS(), iterations=1000)
result = solve(solver, y_vec)

🧪 Running Tests

To verify the installation, run the following command in the project root:

julia --project -e 'using Pkg; Pkg.test()'

🛠 Dependencies

• Optim.jl
• LinearAlgebra

📄 License

MIT License

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### One-Click Deployment (Final Push)
Now that you have the English README and the test file I gave you earlier, run these in your VS Code terminal to finish the job:

```bash
# Save your work
git add .
git commit -m "docs: add English README and comprehensive tests"

# Push to GitHub
git push origin master