# Landau: language for dynamical systems with automatic differentiation

## TL;DR

Landau is a specialized language designed to efficiently compute derivatives of dynamical systems using automatic differentiation, addressing scalability issues in existing tools for practical modeling applications.

## Contribution

It introduces a Turing incomplete, statically typed domain-specific language that enables scalable and optimized automatic differentiation for complex dynamical systems.

## Key findings

- Landau supports complex dynamical equations with minimal effort.
- The language achieves high optimization through source code analysis.
- It effectively handles large numbers of free variables in systems.

## Abstract

Most numerical solvers used to determine free variables of dynamical systems rely on first-order derivatives of the state of the system w.r.t. the free variables. The number of the free variables can be fairly large. One of the approaches of obtaining those derivatives is the integration of the derivatives simultaneously with the dynamical equations, which is best done with the automatic differentiation technique. Even though there exist many automatic differentiation tools, none have been found to be scalable and usable for practical purposes of dynamic systems modeling. Landau is a Turing incomplete statically typed domain-specific language aimed to fill this gap. The Turing incompleteness provides the ability of sophisticated source code analysis and, as a result, a highly optimized compiled code. Among other things, the language syntax supports functions, compile-time ranged for loops, if/else branching constructions, real variables and arrays, and the ability to manually discard calculation where the automatic derivatives values are expected to be negligibly small. In spite of reasonable restrictions, the language is rich enough to express and differentiate any cumbersome paper-equation with practically no effort.