Automatic Differentiation Library

AUTOmatic Derivatives & Jacobians by djmaxus and you

• Functionality
• Motivation
• Project goals
• Anticipated features
• Comparison with autodiff

Functionality

Single variables

```rust use autodj::single::*;

let x : DualF64 = 2.0.into_variable();

// Arithmetic operations are required by trait bounds let _f = x * x + 1.0.into();

// Arithmetic rules itself are defined in Dual trait // on borrowed values for extendability let f = (x*x).add_impl(&1.0.into());

// Dual can be decomposed into a value-derivative pair assert_eq!(f.decompose(), (5.0, 4.0));

// fmt::Display resembles Taylor expansion assert_eq!(format!("{f}"), "5+4∆"); ```

Multiple variables

Multivariate differentiation is based on multiple dual components. Such an approach requires no repetitive and "backward" differentiations. Each partial derivative is tracked separately from the start, and no repetitive calculations are made.

For built-in multivariate specializations, independent variables can be created consistently using .into_variables() method.

Static number of variables

```rust use autodj::array::*;

// consistent set of independent variables let [x, y] : [DualNumber; 2] = [2.0, 3.0].into_variables();

let f = x * (y - 1.0.into());

asserteq!(f.value() , & 4.); asserteq!(f.dual().asref(), &[2., 2.]); asserteq!(format!("{f}") , "4+[2.0, 2.0]∆"); ```

Dynamic number of variables

```rust use autodj::{ fluid::Dual, vector::* }; use std::ops::Add;

let x = vec![1., 2., 3., 4., 5.].into_variables();

let f : DualF64 = x.iter() .map(|x : &DualF64| x.mul_impl(&2.0.into())) .reduce(Add::add) .unwrap();

assert_eq!(f.value(), &30.);

f.dual() .asref() .iter() .foreach(|deriv| assert_eq!(deriv, &2.0) ); ```

Generic dual numbers

rust // A trait with all the behavior defined use autodj::fluid::Dual; // A generic data structure which implements Dual use autodj::solid::DualNumber;

Motivation

I do both academic & business R&D in the area of computational mathematics. As well as many of us, I've written a whole bunch of sophisticated Jacobians by hand.

One day, I learned about automatic differentiation based on dual numbers. Almost the same day, I learned about Rust as well :crab:

Then, I decided to:

• Make it automatic and reliable as much as possible
• Use modern and convenient ecosystem of Rust development

Project goals

• Develop open-source automatic differentiation library for both academic and commercial computational mathematicians
• Gain experience of Rust programming

Anticipated features

You are very welcome to introduce issues to promote most wanted features or to report a bug.

• [x] Generic implementation of dual numbers
• Number of variables to differentiate
  • [x] single
  • multiple
  • [x] static
  • [x] dynamic
  • [x] sparse
  • [ ] Jacobians (efficient layouts in memory to make matrices right away)
• [ ] Named variables (UUID-based)
• [ ] Calculation tracking (partial derivatives of intermediate values)
• Third-party crates support (as features)
  • [x] num-traits
  • [x] linear algebra crates (nalgebra etc.)
• Advanced features
  • [x] Arbitrary number types beside f64
  • [ ] Inter-operability of different dual types (e.g., single and multiple dynamic)
  • [ ] Numerical verification (or replacement) of derivatives (by definition)
  • [ ] Macro for automatic extensions of regular (i.e. non-dual) functions
  • [ ] Optional calculation of derivatives
  • [ ] Backward differentiation probably
  • [ ] Iterator implementation as possible approach to lazy evaluation

Comparison with autodiff

As far as I noticed, autodj currently has the following differences

• Multiple variables out of the box
• fmt::Display for statically-known number of variables
• Left-to-right flow of many operations such as .into-variables(), .eval(), etc.
• Number type is restricted to f64
• No utilization of num and nalgebra crates

Some differences are planned to be eliminated as noted in the roadmap.

Within this crate, you may study & launch test target /tests/autodiff.rs to follow some differences.

shell cargo test --test autodiff -- --show-output