SciML Scientific Machine Learning Documentation and Tutorials

The SciML organization is an opinionated collection of tools for scientific machine learning and differential equation modeling. The organization provides well-maintained tools which compose together as a coherent ecosystem. The following are the relevant resources for users interested in the functionality.

SciML-Wide Documentation

• The SciML Common Interface

• SciMLTutorials

• SciMLBenchmarks

Differential Equations

• DifferentialEquations.jl

• diffeqpy: DifferentialEquations.jl from Python

• diffeqr: DifferentialEquations.jl from R

These resources cover:

• Discrete equations (function maps, discrete stochastic (Gillespie/Markov) simulations)

• Ordinary differential equations (ODEs)

• Split and Partitioned ODEs (Symplectic integrators, IMEX Methods)

• Stochastic ordinary differential equations (SODEs or SDEs)

• Random differential equations (RODEs or RDEs)

• Differential algebraic equations (DAEs)

• Delay differential equations (DDEs)

• Mixed discrete and continuous equations (Hybrid Equations, Jump Diffusions)

• (Stochastic) partial differential equations ((S)PDEs) (with both finite difference and finite element methods)

Partial Differential Equation Modeling and Solving

• MethodOfLines.jl: Finite Difference Methods

• NeuralPDE.jl: Physics-Informed Neural Networks

• NearalOperators.jl: DeepONets and Neural Operator Methods

• DiffEqOperators.jl: Finite Difference Methods

• FEniCS.jl: Finite Element Methods

Core Mathematical Systems

• LinearSolve.jl: High-performance and differentiation-enabled Linear Solvers

• NonlinearSolve.jl: High-performance and differentiation-enabled Nonlinear Solvers

Optimization

• GalacticOptim.jl: Differentiable local and global optimizers all in one interface

Estimation, Inference, and Model Discovery

• DiffEqFlux.jl: Universal and neural differential equations

• NeuralPDE.jl: Physics-Informed Nueral Networks

• DataDrivenDiffEq.jl: Koopman Operator and Symbolic Regression

• DiffEqParamEstim.jl: Easy parameter estimation by maximum likelihood and MAP of differential equations

• DiffEqBayes.jl: Easy Bayesian inference of differential equations

• ReservoirComputing.jl: Reservoir computing methods like echo state networks

Honorable mention to Turing.jl, a probabilistic programming language that composes with the SciML tools.

Surrogate Acceleration

• Surrogates.jl: Fast and differentiable surrogates

Modeling Languages and Domain-Specific Languages

• ModelingToolkit.jl: A Composable Modeling and Simulation Environment

• Catalyst.jl: Fast chemical reaction modeling

• ParameterizedFunctions.jl: Easy definition of differential equation functions

• NBodySimulator.jl: A differentiable N-body problem simulator for molecular dynamics and forcefield equations

Modeling Tools and Primatives

• MultiScaleArrays.jl

• LabelledArrays.jl

• RecursiveArrayTools.jl

Numerical Tools and Primatives

• GlobalSensitivity.jl: Fast and Parallel Global Sensitivity Analysis

• ExponentialUtilities.jl: Fast and GPU matrix exponentials

• Quadrature.jl: Common interface for quadrature and numerical integration

• QuasiMonteCarlo.jl: Quasi-Monte Carlo sampling routines

• SparsityDetection.jl: Automated Jacobian and Hessian sparsity patterns

• PoissonRandom.jl: Fast Poisson random numbers

• AutoOffload.jl: Automatic GPU, TPU, FPGA, Xeon Phi, Multithreaded, Distributed, etc. offloading

Developer Documentation

Please see the developer documentation for information on getting started with developing in the SciML organization. Please see Colprac for the community development practices.

External Tutorials and Teaching Materials

• MIT 18.337J/6.338J: Parallel Computing and Scientific Machine Learning

• MIT 6.S083/18.S190: Computational thinking with Julia + application to the COVID-19 pandemic

• MIT 18.S096 Special Subject in Mathematics: Applications of Scientific Machine Learning

• Various implementations of the classical SIR model in Julia

• Programming for Mathematical Applications