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Top 21 Julia differentialequation Projects

DifferentialEquations.jl
Multilanguage suite for highperformance solvers of differential equations and scientific machine learning (SciML) components. Ordinary differential equations (ODEs), stochastic differential equations (SDEs), delay differential equations (DDEs), differentialalgebraic equations (DAEs), and more in Julia.

ModelingToolkit.jl
An acausal modeling framework for automatically parallelized scientific machine learning (SciML) in Julia. A computer algebra system for integrated symbolics for physicsinformed machine learning and automated transformations of differential equations

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NeuralPDE.jl
PhysicsInformed Neural Networks (PINN) Solvers of (Partial) Differential Equations for Scientific Machine Learning (SciML) accelerated simulation
The documentation has a manifest associated with it: https://docs.sciml.ai/NeuralPDE/dev/#Reproducibility. Instantiating the manifest will give you all of the exact versions used for the documentation build (https://github.com/SciML/NeuralPDE.jl/blob/ghpages/v5.7.0/assets/Manifest.toml). You just ]instantiate folder_of_manifest. Or you can use the Project.toml.

DiffEqFlux.jl
Prebuilt implicit layer architectures with O(1) backprop, GPUs, and stiff+nonstiff DE solvers, demonstrating scientific machine learning (SciML) and physicsinformed machine learning methods

OrdinaryDiffEq.jl
High performance ordinary differential equation (ODE) and differentialalgebraic equation (DAE) solvers, including neural ordinary differential equations (neural ODEs) and scientific machine learning (SciML)
There has been a lot of research in Runge Kutta methods in the last couple decades which resulted in all kind of specialized Runge Kutta methods. You have high order ones, RK methods for stiff problems, embedded RK methods which benefit from adaprive step size control, RKNystrom methods for second order Problems, symplectic RK methods which preserve energy (eg. hamiltonian) ando so on. If you are interested in the numerics and the use cases I highly recommend checking out the Julia Libary OrdinaryDiffEq (https://github.com/SciML/OrdinaryDiffEq.jl). If you look into the documentation you find A LOT of implemented RK methods for all kind of use cases.

Catalyst.jl
Chemical reaction network and systems biology interface for scientific machine learning (SciML). High performance, GPUparallelized, and O(1) solvers in open source software.

DataDrivenDiffEq.jl
Data driven modeling and automated discovery of dynamical systems for the SciML Scientific Machine Learning organization

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SciMLSensitivity.jl
A component of the DiffEq ecosystem for enabling sensitivity analysis for scientific machine learning (SciML). Optimizethendiscretize, discretizethenoptimize, adjoint methods, and more for ODEs, SDEs, DDEs, DAEs, etc.

DiffEqBase.jl
The lightweight Base library for shared types and functionality for defining differential equation and scientific machine learning (SciML) problems


DiffEqGPU.jl
GPUacceleration routines for DifferentialEquations.jl and the broader SciML scientific machine learning ecosystem
Indeed, and this year we created a system for compiling ODE code not just optimized CUDA kernels but also OneAPI kernels, AMD GPU kernels, and Metal. Peer reviewed version is here (https://www.sciencedirect.com/science/article/abs/pii/S00457...), open access is here (https://arxiv.org/abs/2304.06835), and the open source code is at https://github.com/SciML/DiffEqGPU.jl. The key that the paper describes is that in this case kernel generation is about 20x100x faster than PyTorch and Jax (see the Jax compilation in multiple ways in this notebook https://colab.research.google.com/drive/1d7GO5JX31lHbg7jTzz..., extra overhead though from calling Julia from Python but still shows a 10x).
The point really is that while deep learning libraries are amazing, at the end of the day they are DSL and really pull towards one specific way of computing and parallelization. It turns out that way of parallelizing is good for deep learning, but not for all things you may want to accelerate. Sometimes (i.e. cases that aren't dominated by large linear algebra) building problemspecific kernels is a major win, and it's overextrapolating to see ML frameworks do well with GPUs and think that's the only thing that's required. There are many ways to parallelize a code, ML libraries hardcode a very specific way, and it's good for what they are used for but not every problem that can arise.

StochasticDiffEq.jl
Solvers for stochastic differential equations which connect with the scientific machine learning (SciML) ecosystem


NonlinearSolve.jl
Highperformance and differentiationenabled nonlinear solvers (Newton methods), bracketed rootfinding (bisection, Falsi), with sparsity and NewtonKrylov support.


ParameterizedFunctions.jl
A simple domainspecific language (DSL) for defining differential equations for use in scientific machine learning (SciML) and other applications

DiffEqDevTools.jl
Benchmarking, testing, and development tools for differential equations and scientific machine learning (SciML)


SimpleDiffEq.jl
Simple differential equation solvers in native Julia for scientific machine learning (SciML)

SciPyDiffEq.jl
Wrappers for the SciPy differential equation solvers for the SciML Scientific Machine Learning organization
Project mention: SciPy: Interested in adopting PRIMA, but little appetite for more Fortran code  news.ycombinator.com  20230518Interesting response. I develop the Julia SciML organization https://sciml.ai/ and we'd be more than happy to work with you to get wrappers for PRIMA into Optimization.jl's general interface (https://docs.sciml.ai/Optimization/stable/). Please get in touch and we can figure out how to set this all up. I personally would be curious to try this out and do some benchmarks against nlopt methods.

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Julia differentialequations related posts
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 Why Fortran is a scientific powerhouse
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Index
What are some of the best opensource differentialequation projects in Julia? This list will help you:
Project  Stars  

1  DifferentialEquations.jl  2,746 
2  ModelingToolkit.jl  1,330 
3  NeuralPDE.jl  899 
4  DiffEqFlux.jl  833 
5  OrdinaryDiffEq.jl  499 
6  Catalyst.jl  419 
7  DataDrivenDiffEq.jl  398 
8  Surrogates.jl  312 
9  SciMLSensitivity.jl  306 
10  DiffEqBase.jl  295 
11  ComponentArrays.jl  275 
12  DiffEqGPU.jl  267 
13  StochasticDiffEq.jl  233 
14  ReservoirComputing.jl  197 
15  NonlinearSolve.jl  191 
16  ModelingToolkitStandardLibrary.jl  97 
17  ParameterizedFunctions.jl  76 
18  DiffEqDevTools.jl  46 
19  BoundaryValueDiffEq.jl  39 
20  SimpleDiffEq.jl  22 
21  SciPyDiffEq.jl  20 