SciML/ModelingToolkit.jlPublic

NotificationsYou must be signed in to change notification settings
Fork234
Star1.6k

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

License

View license

1.6k stars 234 forks Branches Tags Activity

Star

Notifications

You must be signed in to change notification settings

Branches Tags

Folders and files

Name		Name	Last commit message	Last commit date
Latest commit History 9,139 Commits
.github		.github
benchmark		benchmark
docs		docs
ext		ext
src		src
test		test
.JuliaFormatter.toml		.JuliaFormatter.toml
.codecov.yml		.codecov.yml
.gitignore		.gitignore
.typos.toml		.typos.toml
CITATION.bib		CITATION.bib
CONTRIBUTING.md		CONTRIBUTING.md
LICENSE.md		LICENSE.md
NEWS.md		NEWS.md
Project.toml		Project.toml
README.md		README.md
demo.jl		demo.jl

Repository files navigation

ModelingToolkit.jl

ModelingToolkit.jl is a modeling framework for high-performance symbolic-numeric computationin scientific computing and scientific machine learning.It allows for users to give a high-level description of a model forsymbolic preprocessing to analyze and enhance the model. ModelingToolkit canautomatically generate fast functions for model components like Jacobiansand Hessians, along with automatically sparsifying and parallelizing thecomputations. Automatic transformations, such as index reduction, can be appliedto the model to make it easier for numerical solvers to handle.

For information on using the package,see the stable documentation. Use thein-development documentation for the version ofthe documentation which contains the unreleased features.

Standard Library

For a standard library of ModelingToolkit components and blocks, check out theModelingToolkitStandardLibrary

High-Level Examples

First, let's define a second order riff on the Lorenz equations, symbolicallylower it to a first order system, symbolically generate the Jacobian functionfor the numerical integrator, and solve it.

using ModelingToolkitusing ModelingToolkit: t_nounits as t, D_nounits as D# Defines a ModelingToolkit `System` model.@parameters σ ρ β@variablesx(t)y(t)z(t)eqs= [D(D(x))~ σ* (y- x),D(y)~ x* (ρ- z)- y,D(z)~ x* y- β* z]@mtkcompile sys=System(eqs, t)# Simulate the model for a specific condition (initial condition and parameter values).using OrdinaryDiffEqDefaultsim_cond= [D(x)=>2.0,    x=>1.0,    y=>0.0,    z=>0.0,    σ=>28.0,    ρ=>10.0,    β=>8/3]tend=100.0prob=ODEProblem(sys, sim_cond, tend; jac=true)sol=solve(prob)# Plot the solution in phase-space.using Plotsplot(sol, idxs= (x, y))

This will have automatically generated fast Jacobian functions, makingit more optimized than directly building a function. In addition, we can thenuse ModelingToolkit to compose multiple ODE subsystems. Now, let's define twointeracting Lorenz equations and simulate the resulting Differential-AlgebraicEquation (DAE):

using ModelingToolkitusing ModelingToolkit: t_nounits as t, D_nounits as D# Defines two lorenz system models.eqs= [D(x)~ σ* (y- x),D(y)~ x* (ρ- z)- y,D(z)~ x* y- β* z]@named lorenz1=System(eqs, t)@named lorenz2=System(eqs, t)# Connect the two models, creating a single model.@variablesa(t)@parameters γconnections= [0~ lorenz1.x+ lorenz2.y+ a* γ]@mtkcompile connected_lorenz=System(connections, t; systems= [lorenz1, lorenz2])# Simulate the model for a specific condition (initial condition and parameter values).using OrdinaryDiffEqDefaultsim_cond= [    lorenz1.x=>1.0,    lorenz1.y=>0.0,    lorenz1.z=>0.0,    lorenz2.x=>0.0,    lorenz2.z=>0.0,    a=>2.0,    lorenz1.σ=>10.0,    lorenz1.ρ=>28.0,    lorenz1.β=>8/3,    lorenz2.σ=>10.0,    lorenz2.ρ=>28.0,    lorenz2.β=>8/3,    γ=>2.0]tend=100.0prob=ODEProblem(connected_lorenz, sim_cond, tend)sol=solve(prob)# Plot the solution in phase-space.using Plotsplot(sol, idxs= (a, lorenz1.x, lorenz2.z))

Citation

If you use ModelingToolkit.jl in your research, please citethis paper:

@misc{ma2021modelingtoolkit,      title={ModelingToolkit: A Composable Graph Transformation System For Equation-Based Modeling},      author={Yingbo Ma and Shashi Gowda and Ranjan Anantharaman and Chris Laughman and Viral Shah and Chris Rackauckas},      year={2021},      eprint={2103.05244},      archivePrefix={arXiv},      primaryClass={cs.MS}}