ModelingToolkit.jl: High-Performance Symbolic-Numeric Equation-Based Modeling

ModelingToolkit.jl is a modeling language for high-performance symbolic-numeric computation in scientific computing and scientific machine learning. It then mixes ideas from symbolic computational algebra systems with causal and acausal equation-based modeling frameworks to give an extendable and parallel modeling system. It allows for users to give a high-level description of a model for symbolic preprocessing to analyze and enhance the model. Automatic transformations, such as index reduction, can be applied to the model before solving in order to make it easily handle equations would could not be solved when modeled without symbolic intervention.


To install ModelingToolkit.jl, use the Julia package manager:

using Pkg


If you use ModelingToolkit in your work, please cite the following:

      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},

Feature Summary

ModelingToolkit.jl is a symbolic-numeric modeling package. Thus it combines some of the features from symbolic computing packages like SymPy or Mathematica with the ideas of equation-based modeling systems like the causal Simulink and the acausal Modelica. It bridges the gap between many different kinds of equations, allowing one to quickly and easily transform systems of DAEs into optimization problems, or vice-versa, and then simplify and parallelize the resulting expressions before generating code.

Feature List

  • Causal and acausal modeling (Simulink/Modelica)
  • Automated model transformation, simplification, and composition
  • Automatic conversion of numerical models into symbolic models
  • Composition of models through the components, a lazy connection system, and tools for expanding/flattening
  • Pervasive parallelism in symbolic computations and generated functions
  • Transformations like alias elimination and tearing of nonlinear systems for efficiently numerically handling large-scale systems of equations
  • The ability to use the entire Symbolics.jl Computer Algebra System (CAS) as part of the modeling process.
  • Import models from common formats like SBML, CellML, BioNetGen, and more.
  • Extendability: the whole system is written in pure Julia, so adding new functions, simplification rules, and model transformations has no barrier.

For information on how to use the Symbolics.jl CAS system that ModelingToolkit.jl is built on, consult the Symbolics.jl documentation

Equation Types

  • Ordinary differential equations
  • Stochastic differential equations
  • Partial differential equations
  • Nonlinear systems
  • Optimization problems
  • Continuous-Time Markov Chains
  • Chemical Reactions (via Catalyst.jl)
  • Nonlinear Optimal Control

Standard Library

For quick development, ModelingToolkit.jl includes ModelingToolkitStandardLibrary.jl, a standard library of prebuilt components for the ModelingToolkit ecosystem.

Model Import Formats

  • CellMLToolkit.jl: Import CellML models into ModelingToolkit
    • Repository of more than a thousand pre-made models
    • Focus on biomedical models in areas such as: Calcium Dynamics, Cardiovascular Circulation, Cell Cycle, Cell Migration, Circadian Rhythms, Electrophysiology, Endocrine, Excitation-Contraction Coupling, Gene Regulation, Hepatology, Immunology, Ion Transport, Mechanical Constitutive Laws, Metabolism, Myofilament Mechanics, Neurobiology, pH Regulation, PKPD, Protein Modules, Signal Transduction, and Synthetic Biology.
  • SBMLToolkit.jl: Import SBML models into ModelingToolkit
    • Uses the robust libsbml library for parsing and transforming the SBML
  • ReactionNetworkImporters.jl: Import various models into ModelingToolkit
    • Supports the BioNetGen .net file
    • Supports importing networks specified by stoichiometric matrices

Extension Libraries

Because ModelingToolkit.jl is the core foundation of a equation-based modeling ecosystem, there is a large set of libraries adding features to this system. Below is an incomplete list of extension libraries one may want to be aware of:

  • Catalyst.jl: Symbolic representations of chemical reactions
    • Symbolically build and represent large systems of chemical reactions
    • Generate code for ODEs, SDEs, continuous-time Markov Chains, and more
    • Simulate the models using the SciML ecosystem with O(1) Gillespie methods
  • DataDrivenDiffEq.jl: Automatic identification of equations from data
    • Automated construction of ODEs and DAEs from data
    • Representations of Koopman operators and Dynamic Mode Decomposition (DMD)
  • MomentClosure.jl: Automatic transformation of ReactionSystems into deterministic systems
    • Generates ODESystems for the moment closures
    • Allows for geometrically-distributed random reaction rates
  • ReactionMechanismSimulator.jl: Simulating and analyzing large chemical reaction mechanisms
    • Ideal gas and dilute liquid phases.
    • Constant T and P and constant V adiabatic ideal gas reactors.
    • Constant T and V dilute liquid reactors.
    • Diffusion limited rates. Sensitivity analysis for all reactors.
    • Flux diagrams with molecular images (if molecular information is provided).
  • NumCME.jl: High-performance simulation of chemical master equations (CME)
    • Transient solution of the CME
    • Dynamic state spaces
    • Accepts reaction systems defined using Catalyst.jl DSL.
  • FiniteStateProjection.jl: High-performance simulation of chemical master equations (CME) via finite state projections
    • Accepts reaction systems defined using Catalyst.jl DSL.

Compatible Numerical Solvers

All of the symbolic systems have a direct conversion to a numerical system which can then be handled through the SciML interfaces. For example, after building a model and performing symbolic manipulations, an ODESystem can be converted into an ODEProblem to then be solved by a numerical ODE solver. Below is a list of the solver libraries which are the numerical targets of the ModelingToolkit system:

  • DifferentialEquations.jl
    • Multi-package interface of high performance numerical solvers for ODESystem, SDESystem, and JumpSystem
  • NonlinearSolve.jl
    • High performance numerical solving of NonlinearSystem
  • Optimization.jl
    • Multi-package interface for numerical solving OptimizationSystem
  • NeuralPDE.jl
    • Physics-Informed Neural Network (PINN) training on PDESystem
  • MethodOfLines.jl
    • Automated finite difference method (FDM) discretization of PDESystem



The documentation of this SciML package was built using these direct dependencies,
Status `~/work/ModelingToolkit.jl/ModelingToolkit.jl/docs/Project.toml`
  [6e4b80f9] BenchmarkTools v1.3.2
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  [37e2e46d] LinearAlgebra
and using this machine and Julia version.
Julia Version 1.8.3
Commit 0434deb161e (2022-11-14 20:14 UTC)
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 2 × Intel(R) Xeon(R) CPU E5-2673 v3 @ 2.40GHz
  LIBM: libopenlibm
  LLVM: libLLVM-13.0.1 (ORCJIT, haswell)
  Threads: 1 on 2 virtual cores
A more complete overview of all dependencies and their versions is also provided.
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  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [b27032c2] LibCURL v0.6.3
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.8.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.0
  [a4e569a6] Tar v1.10.1
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v0.5.2+0
  [deac9b47] LibCURL_jll v7.84.0+0
  [29816b5a] LibSSH2_jll v1.10.2+0
  [c8ffd9c3] MbedTLS_jll v2.28.0+0
  [14a3606d] MozillaCACerts_jll v2022.2.1
  [4536629a] OpenBLAS_jll v0.3.20+0
  [05823500] OpenLibm_jll v0.8.1+0
  [efcefdf7] PCRE2_jll v10.40.0+0
  [bea87d4a] SuiteSparse_jll v5.10.1+0
  [83775a58] Zlib_jll v1.2.12+3
  [8e850b90] libblastrampoline_jll v5.1.1+0
  [8e850ede] nghttp2_jll v1.48.0+0
  [3f19e933] p7zip_jll v17.4.0+0
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`
You can also download the manifest file and the project file.