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Tutorials for doing scientific machine learning (SciML) and high-performance differential equation solving with open source software.

DifferentialEquations.jl

Multi-language suite for high-performance solvers of differential equations and scientific machine learning (SciML) components. Ordinary differential equations (ODEs), stochastic differential equations (SDEs), delay differential equations (DDEs), differential-algebraic equations (DAEs), and more in Julia.

Julia

2,599

SciMLBook

Parallel Computing and Scientific Machine Learning (SciML): Methods and Applications (MIT 18.337J/6.338J)

HTML

1,841

ModelingToolkit.jl

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

Julia

1,424

NeuralPDE.jl

Physics-Informed Neural Networks (PINN) Solvers of (Partial) Differential Equations for Scientific Machine Learning (SciML) accelerated simulation

Julia

961

DiffEqFlux.jl

Pre-built implicit layer architectures with O(1) backprop, GPUs, and stiff+non-stiff DE solvers, demonstrating scientific machine learning (SciML) and physics-informed machine learning methods

Julia

858

SciMLTutorials.jl

Tutorials for doing scientific machine learning (SciML) and high-performance differential equation solving with open source software.

CSS

710

Optimization.jl

Mathematical Optimization in Julia. Local, global, gradient-based and derivative-free. Linear, Quadratic, Convex, Mixed-Integer, and Nonlinear Optimization in one simple, fast, and differentiable interface.

Julia

708

OrdinaryDiffEq.jl

High performance ordinary differential equation (ODE) and differential-algebraic equation (DAE) solvers, including neural ordinary differential equations (neural ODEs) and scientific machine learning (SciML)

Julia

528

diffeqpy

Solving differential equations in Python using DifferentialEquations.jl and the SciML Scientific Machine Learning organization

Python

521

Catalyst.jl

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

Julia

462

DataDrivenDiffEq.jl

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

Julia

405

SciMLSensitivity.jl

A component of the DiffEq ecosystem for enabling sensitivity analysis for scientific machine learning (SciML). Optimize-then-discretize, discretize-then-optimize, adjoint methods, and more for ODEs, SDEs, DDEs, DAEs, etc.

Julia

330

Surrogates.jl

Surrogate modeling and optimization for scientific machine learning (SciML)

Julia

328

SciMLBenchmarks.jl

Scientific machine learning (SciML) benchmarks, AI for science, and (differential) equation solvers. Covers Julia, Python (PyTorch, Jax), MATLAB, R

MATLAB

299

DiffEqOperators.jl

Linear operators for discretizations of differential equations and scientific machine learning (SciML)

Julia

285

DiffEqGPU.jl

GPU-acceleration routines for DifferentialEquations.jl and the broader SciML scientific machine learning ecosystem

Julia

283

FluxNeuralOperators.jl

DeepONets, (Fourier) Neural Operators, Physics-Informed Neural Operators, and more in Julia

Julia

267

DiffEqDocs.jl

Documentation for the DiffEq differential equations and scientific machine learning (SciML) ecosystem

Julia

262

DiffEqBase.jl

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

Julia

254

LinearSolve.jl

LinearSolve.jl: High-Performance Unified Interface for Linear Solvers in Julia. Easily switch between factorization and Krylov methods, add preconditioners, and all in one interface.

Julia

245

Integrals.jl

A common interface for quadrature and numerical integration for the SciML scientific machine learning organization

Julia

226

NonlinearSolve.jl

High-performance and differentiation-enabled nonlinear solvers (Newton methods), bracketed rootfinding (bisection, Falsi), with sparsity and Newton-Krylov support.

Julia

226

DataInterpolations.jl

A library of data interpolation and smoothing functions

Julia

212

SciMLStyle

A style guide for stylish Julia developers

Julia

211

StochasticDiffEq.jl

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

Julia

209

ReservoirComputing.jl

Reservoir computing utilities for scientific machine learning (SciML)

Julia

206

Sundials.jl

Julia interface to Sundials, including a nonlinear solver (KINSOL), ODE's (CVODE and ARKODE), and DAE's (IDA) in a SciML scientific machine learning enabled manner

Julia

195

RecursiveArrayTools.jl

Tools for easily handling objects like arrays of arrays and deeper nestings in scientific machine learning (SciML) and other applications

Julia

167

MethodOfLines.jl

Automatic Finite Difference PDE solving with Julia SciML

Julia

160

diffeqr

Solving differential equations in R using DifferentialEquations.jl and the SciML Scientific Machine Learning ecosystem

140

JumpProcesses.jl

Build and simulate jump equations like Gillespie simulations and jump diffusions with constant and state-dependent rates and mix with differential equations and scientific machine learning (SciML)

Julia

140

SciMLBase.jl

The Base interface of the SciML ecosystem

Julia

129

NBodySimulator.jl

A differentiable simulator for scientific machine learning (SciML) with N-body problems, including astrophysical and molecular dynamics

Julia

128

ColPrac

Contributor's Guide on Collaborative Practices for Community Packages

Julia

123

DiffEqBayes.jl

Extension functionality which uses Stan.jl, DynamicHMC.jl, and Turing.jl to estimate the parameters to differential equations and perform Bayesian probabilistic scientific machine learning

Julia

121

LabelledArrays.jl

Arrays which also have a label for each element for easy scientific machine learning (SciML)

Julia

120

PolyChaos.jl

A Julia package to construct orthogonal polynomials, their quadrature rules, and use it with polynomial chaos expansions.

Julia

116

SymbolicNumericIntegration.jl

SymbolicNumericIntegration.jl: Symbolic-Numerics for Solving Integrals

Julia

116

ModelingToolkitStandardLibrary.jl

A standard library of components to model the world and beyond

Julia

112

PreallocationTools.jl

Tools for building non-allocating pre-cached functions in Julia, allowing for GC-free usage of automatic differentiation in complex codes

Julia

111

StructuralIdentifiability.jl

Fast and automatic structural identifiability software for ODE systems

Julia

110

ODE.jl

Assorted basic Ordinary Differential Equation solvers for scientific machine learning (SciML). Deprecated: Use DifferentialEquations.jl instead.

Julia

103

QuasiMonteCarlo.jl

Lightweight and easy generation of quasi-Monte Carlo sequences with a ton of different methods on one API for easy parameter exploration in scientific machine learning (SciML)

Julia

102

RuntimeGeneratedFunctions.jl

Functions generated at runtime without world-age issues or overhead

Julia

100

FEniCS.jl

A scientific machine learning (SciML) wrapper for the FEniCS Finite Element library in the Julia programming language

Julia

DiffEqCallbacks.jl

A library of useful callbacks for hybrid scientific machine learning (SciML) with augmented differential equation solvers

Julia

ExponentialUtilities.jl

Fast and differentiable implementations of matrix exponentials, Krylov exponential matrix-vector multiplications ("expmv"), KIOPS, ExpoKit functions, and more. All your exponential needs in SciML form.

Julia

EllipsisNotation.jl

Julia-based implementation of ellipsis array indexing notation `..`

Julia

EasyModelAnalysis.jl

High level functions for analyzing the output of simulations

Julia

AutoOptimize.jl

Automatic optimization and parallelization for Scientific Machine Learning (SciML)

Julia

ParameterizedFunctions.jl

A simple domain-specific language (DSL) for defining differential equations for use in scientific machine learning (SciML) and other applications

Julia

HighDimPDE.jl

A Julia package for Deep Backwards Stochastic Differential Equation (Deep BSDE) and Feynman-Kac methods to solve high-dimensional PDEs without the curse of dimensionality

Julia

SciMLExpectations.jl

Fast uncertainty quantification for scientific machine learning (SciML) and differential equations

Julia

MultiScaleArrays.jl

A framework for developing multi-scale arrays for use in scientific machine learning (SciML) simulations

Julia

SimpleNonlinearSolve.jl

Fast and simple nonlinear solvers for the SciML common interface. Newton, Broyden, Bisection, Falsi, and more rootfinders on a standard interface.

Julia

DiffEqNoiseProcess.jl

A library of noise processes for stochastic systems like stochastic differential equations (SDEs) and other systems that are present in scientific machine learning (SciML)

Julia

CellMLToolkit.jl

CellMLToolkit.jl is a Julia library that connects CellML models to the Scientific Julia ecosystem.

Julia

SciMLDocs

Global documentation for the Julia SciML Scientific Machine Learning Organization

Julia

SparsityDetection.jl

Automatic detection of sparsity in pure Julia functions for sparsity-enabled scientific machine learning (SciML)

Julia

DelayDiffEq.jl

Delay differential equation (DDE) solvers in Julia for the SciML scientific machine learning ecosystem. Covers neutral and retarded delay differential equations, and differential-algebraic equations.

Julia

DiffEqProblemLibrary.jl

A library of premade problems for examples and testing differential equation solvers and other SciML scientific machine learning tools

Julia

sciml.ai

The SciML Scientific Machine Learning Software Organization Website

CSS

DiffEqParamEstim.jl

Easy scientific machine learning (SciML) parameter estimation with pre-built loss functions

Julia

Static.jl

Static types useful for dispatch and generated functions.

Julia

GlobalSensitivity.jl

Robust, Fast, and Parallel Global Sensitivity Analysis (GSA) in Julia

Julia

DeepEquilibriumNetworks.jl

Implicit Layer Machine Learning via Deep Equilibrium Networks, O(1) backpropagation with accelerated convergence.

Julia

MinimallyDisruptiveCurves.jl

Finds relationships between the parameters of a mathematical model

Julia

DiffEqPhysics.jl

A library for building differential equations arising from physical problems for physics-informed and scientific machine learning (SciML)

Julia

OperatorLearning.jl

No need to train, he's a smooth operator

Julia

MuladdMacro.jl

This package contains a macro for converting expressions to use muladd calls and fused-multiply-add (FMA) operations for high-performance in the SciML scientific machine learning ecosystem

Julia

DiffEqDevTools.jl

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

Julia

BoundaryValueDiffEq.jl

Boundary value problem (BVP) solvers for scientific machine learning (SciML)

Julia

SciMLOperators.jl

SciMLOperators.jl: Matrix-Free Operators for the SciML Scientific Machine Learning Common Interface in Julia

Julia

SBMLToolkit.jl

SBML differential equation and chemical reaction model (Gillespie simulations) for Julia's SciML ModelingToolkit

Julia

HelicopterSciML.jl

Helicopter Scientific Machine Learning (SciML) Challenge Problem

Julia

ADTypes.jl

Repository for automatic differentiation backend types

Julia

RootedTrees.jl

A collection of functionality around rooted trees to generate order conditions for Runge-Kutta methods in Julia for differential equations and scientific machine learning (SciML)

Julia

SciMLWorkshop.jl

Workshop materials for training in scientific computing and scientific machine learning

Julia

AutoOffload.jl

Automatic GPU, TPU, FPGA, Xeon Phi, Multithreaded, Distributed, etc. offloading for scientific machine learning (SciML) and differential equations

Julia

ModelOrderReduction.jl

High-level model-order reduction to automate the acceleration of large-scale simulations

Julia

ModelingToolkitCourse

A course on composable system modeling, differential-algebraic equations, acausal modeling, compilers for simulation, and building digital twins of real-world devices

Julia

DifferenceEquations.jl

Solving difference equations with DifferenceEquations.jl and the SciML ecosystem.

Julia

DASSL.jl

Solves stiff differential algebraic equations (DAE) using variable stepsize backwards finite difference formula (BDF) in the SciML scientific machine learning organization

Julia

FiniteVolumeMethod.jl

Solver for two-dimensional conservation equations using the finite volume method in Julia.

Julia

SteadyStateDiffEq.jl

Solvers for steady states in scientific machine learning (SciML)

Julia

TruncatedStacktraces.jl

Simpler stacktraces for the Julia Programming Language

Julia

PDESystemLibrary.jl

A library of systems of partial differential equations, as defined with ModelingToolkit.jl in Julia

Julia

DiffEqOnline

It's Angular2 business in the front, and a Julia party in the back! It's scientific machine learning (SciML) for the web

TypeScript

ReactionNetworkImporters.jl

Julia Catalyst.jl importers for various reaction network file formats like BioNetGen and stoichiometry matrices

Julia

StochasticDelayDiffEq.jl

Stochastic delay differential equations (SDDE) solvers for the SciML scientific machine learning ecosystem

Julia

DiffEqOnlineServer

Backend for DiffEqOnline, a webapp for scientific machine learning (SciML)

Julia

MathML.jl

Julia MathML parser

Julia

IRKGaussLegendre.jl

Implicit Runge-Kutta Gauss-Legendre 16th order (Julia)

Jupyter Notebook

SimpleDiffEq.jl

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

Julia

DiffEqFinancial.jl

Differential equation problem specifications and scientific machine learning for common financial models

Julia

ModelingToolkitNeuralNets.jl

Symbolic-Numeric Universal Differential Equations for Automating Scientific Machine Learning (SciML)

Julia

SciPyDiffEq.jl

Wrappers for the SciPy differential equation solvers for the SciML Scientific Machine Learning organization

Julia

OptimalControl.jl

A component of the SciML scientific machine learning ecosystem for optimal control

Julia

MATLABDiffEq.jl

Common interface bindings for the MATLAB ODE solvers via MATLAB.jl for the SciML Scientific Machine Learning ecosystem

Julia

100

IfElse.jl

Under some conditions you may need this function

Julia

SciML/SciMLTutorialsOutput

SciML

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