DifferentialEquations.jl – A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia

TLDR

DifferentialEquations.jl is a Julia package for solving differential equations. DifferentialEquations.jl achieves a unified, high‑performance interface by leveraging Julia’s multiple dispatch, metaprogramming, plot recipes, FFI, and call‑overloading, while integrating features such as arbitrary number systems, physical units, multithreading, parallelism, symbolic Jacobian calculation, and an algorithm testing and benchmarking suite. The package supports a wide range of equation types—including discrete, ODEs, SDEs, DAEs, DDEs, hybrid, jump diffusions, and PDEs—while delivering high performance and extensive features. The work was funded by NIH, NSF, and other agencies, including the NSF Graduate Research Fellowship, National Academies, Ford Foundation, and NIH Award T32 EB009418.

Abstract

DifferentialEquations.jl is a package for solving differential equations in Julia. It covers discrete equations (function maps, discrete stochastic (Gillespie/Markov) simulations), ordinary differential equations, stochastic differential equations, algebraic differential equations, delay differential equations, hybrid differential equations, jump diffusions, and (stochastic) partial differential equations. Through extensive use of multiple dispatch, metaprogramming, plot recipes, foreign function interfaces (FFI), and call-overloading, DifferentialEquations.jl offers a unified user interface to solve and analyze various forms of differential equations while not sacrificing features or performance. Many modern features are integrated into the solvers, such as allowing arbitrary user-defined number systems for high-precision and arithmetic with physical units, built-in multithreading and parallelism, and symbolic calculation of Jacobians. Integrated into the package is an algorithm testing and benchmarking suite to both ensure accuracy and serve as an easy way for researchers to develop and distribute their own methods. Together, these features build a highly extendable suite which is feature-rich and highly performant.Funding statement: This work was partially supported by NIH grants P50GM76516 and R01GM107264 and NSF grants DMS1562176 and DMS1161621. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1321846, the National Academies of Science, Engineering, and Medicine via the Ford Foundation, and the National Institutes of Health Award T32 EB009418. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the NIH.