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 symbolic transformations, such as index reduction of differential-algebraic equations, make it possible to solve equations that are impossible to solve with a purely numeric-based technique.

Installation

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

using Pkg
Pkg.add("ModelingToolkit")

Citation

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

@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}
}

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.
Extensibility: 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 an 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

Contributing

Please refer to the SciML ColPrac: Contributor's Guide on Collaborative Practices for Community Packages for guidance on PRs, issues, and other matters relating to contributing to SciML.
See the SciML Style Guide for common coding practices and other style decisions.
There are a few community forums:
- The #diffeq-bridged and #sciml-bridged channels in the Julia Slack
- The #diffeq-bridged and #sciml-bridged channels in the Julia Zulip
- On the Julia Discourse forums
- See also SciML Community page

Reproducibility

The documentation of this SciML package was built using these direct dependencies,

Status `~/work/ModelingToolkit.jl/ModelingToolkit.jl/docs/Project.toml`
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Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`

and using this machine and Julia version.

Julia Version 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver3)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_DEBUG = Documenter

A more complete overview of all dependencies and their versions is also provided.

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  [b3c3ace0] RangeArrays v0.3.2
  [c84ed2f1] Ratios v0.4.5
  [3cdcf5f2] RecipesBase v1.3.4
  [01d81517] RecipesPipeline v0.6.12
  [731186ca] RecursiveArrayTools v3.32.1
  [189a3867] Reexport v1.2.2
  [2792f1a3] RegistryInstances v0.1.0
  [05181044] RelocatableFolders v1.0.1
  [ae029012] Requires v1.3.1
  [ae5879a3] ResettableStacks v1.1.1
  [79098fc4] Rmath v0.8.0
  [21fd56a4] RobustAndOptimalControl v0.4.38
  [f2b01f46] Roots v2.2.7
  [5eaf0fd0] RoundingEmulator v0.2.1
  [7e49a35a] RuntimeGeneratedFunctions v0.5.14
  [9dfe8606] SCCNonlinearSolve v1.1.0
  [fdea26ae] SIMD v3.7.1
  [94e857df] SIMDTypes v0.1.0
  [476501e8] SLEEFPirates v0.6.43
  [0bca4576] SciMLBase v2.85.0
  [19f34311] SciMLJacobianOperators v0.1.3
  [c0aeaf25] SciMLOperators v0.3.13
  [53ae85a6] SciMLStructures v1.7.0
  [6c6a2e73] Scratch v1.2.1
  [efcf1570] Setfield v1.1.2
  [65257c39] ShaderAbstractions v0.5.0
  [992d4aef] Showoff v1.0.3
  [73760f76] SignedDistanceFields v0.4.0
  [777ac1f9] SimpleBufferStream v1.2.0
  [727e6d20] SimpleNonlinearSolve v2.3.0
  [699a6c99] SimpleTraits v0.9.4
  [ce78b400] SimpleUnPack v1.1.0
  [45858cf5] Sixel v0.1.3
  [a2af1166] SortingAlgorithms v1.2.1
  [9f842d2f] SparseConnectivityTracer v0.6.17
  [47a9eef4] SparseDiffTools v2.25.0
  [0a514795] SparseMatrixColorings v0.4.18
  [d4ead438] SpatialIndexing v0.1.6
  [276daf66] SpecialFunctions v2.5.1
  [860ef19b] StableRNGs v1.0.2
  [cae243ae] StackViews v0.1.1
  [40b095a5] StateSpaceSets v2.4.0
  [aedffcd0] Static v1.2.0
  [0d7ed370] StaticArrayInterface v1.8.0
  [90137ffa] StaticArrays v1.9.13
  [1e83bf80] StaticArraysCore v1.4.3
  [82ae8749] StatsAPI v1.7.0
  [2913bbd2] StatsBase v0.34.4
  [4c63d2b9] StatsFuns v1.4.0
  [789caeaf] StochasticDiffEq v6.76.0
  [7792a7ef] StrideArraysCore v0.5.7
  [892a3eda] StringManipulation v0.4.1
  [09ab397b] StructArrays v0.7.1
  [2efcf032] SymbolicIndexingInterface v0.3.39
  [19f23fe9] SymbolicLimits v0.2.2
  [d1185830] SymbolicUtils v3.27.0
  [0c5d862f] Symbolics v6.38.0
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.12.0
  [ed4db957] TaskLocalValues v0.1.2
  [62fd8b95] TensorCore v0.1.1
  [8ea1fca8] TermInterface v2.0.0
  [5d786b92] TerminalLoggers v0.1.7
  [1c621080] TestItems v1.0.0
  [8290d209] ThreadingUtilities v0.5.3
  [731e570b] TiffImages v0.11.3
  [a759f4b9] TimerOutputs v0.5.28
  [0796e94c] Tokenize v0.5.29
  [3bb67fe8] TranscodingStreams v0.11.3
  [410a4b4d] Tricks v0.1.10
  [981d1d27] TriplotBase v0.1.0
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.5.2
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.22.0
  [45397f5d] UnitfulLatexify v1.6.4
  [a7c27f48] Unityper v0.1.6
  [41fe7b60] Unzip v0.2.0
  [409d34a3] VectorInterface v0.5.0
  [3d5dd08c] VectorizationBase v0.21.71
  [19fa3120] VertexSafeGraphs v0.2.0
  [29a6e085] Wavelets v0.10.1
  [897b6980] WeakValueDicts v0.1.0
  [e3aaa7dc] WebP v0.1.3
  [efce3f68] WoodburyMatrices v1.0.0
  [a5390f91] ZipFile v0.10.1
⌅ [68821587] Arpack_jll v3.5.1+1
  [6e34b625] Bzip2_jll v1.0.9+0
  [4e9b3aee] CRlibm_jll v1.0.1+0
⌃ [83423d85] Cairo_jll v1.18.4+0
  [ee1fde0b] Dbus_jll v1.16.2+0
  [5ae413db] EarCut_jll v2.2.4+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
  [2e619515] Expat_jll v2.6.5+0
⌅ [b22a6f82] FFMPEG_jll v4.4.4+1
  [f5851436] FFTW_jll v3.3.11+0
  [a3f928ae] Fontconfig_jll v2.16.0+0
  [d7e528f0] FreeType2_jll v2.13.4+0
  [559328eb] FriBidi_jll v1.0.17+0
  [0656b61e] GLFW_jll v3.4.0+2
  [d2c73de3] GR_jll v0.73.14+0
  [78b55507] Gettext_jll v0.21.0+0
  [59f7168a] Giflib_jll v5.2.3+0
  [f8c6e375] Git_jll v2.49.0+0
⌃ [7746bdde] Glib_jll v2.82.4+0
  [3b182d85] Graphite2_jll v1.3.15+0
  [0234f1f7] HDF5_jll v1.14.6+0
  [2e76f6c2] HarfBuzz_jll v8.5.0+0
  [e33a78d0] Hwloc_jll v2.12.0+0
  [905a6f67] Imath_jll v3.1.11+0
  [1d5cc7b8] IntelOpenMP_jll v2025.0.4+0
  [aacddb02] JpegTurbo_jll v3.1.1+0
  [c1c5ebd0] LAME_jll v3.100.2+0
  [88015f11] LERC_jll v4.0.1+0
  [1d63c593] LLVMOpenMP_jll v18.1.8+0
  [dd4b983a] LZO_jll v2.10.3+0
  [81d17ec3] L_BFGS_B_jll v3.0.1+0
⌅ [e9f186c6] Libffi_jll v3.2.2+2
  [7e76a0d4] Libglvnd_jll v1.7.1+1
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.41.0+0
  [89763e89] Libtiff_jll v4.7.1+0
  [38a345b3] Libuuid_jll v2.41.0+0
  [856f044c] MKL_jll v2025.0.1+1
  [7cb0a576] MPICH_jll v4.3.0+1
  [f1f71cc9] MPItrampoline_jll v5.5.3+0
  [9237b28f] MicrosoftMPI_jll v10.1.4+3
  [e7412a2a] Ogg_jll v1.3.5+1
  [6cdc7f73] OpenBLASConsistentFPCSR_jll v0.3.29+0
  [18a262bb] OpenEXR_jll v3.2.4+0
  [fe0851c0] OpenMPI_jll v5.0.7+2
  [458c3c95] OpenSSL_jll v3.5.0+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.3.3+0
  [36c8627f] Pango_jll v1.56.1+0
  [30392449] Pixman_jll v0.44.2+0
⌅ [c0090381] Qt6Base_jll v6.7.1+1
⌅ [629bc702] Qt6Declarative_jll v6.7.1+2
⌅ [ce943373] Qt6ShaderTools_jll v6.7.1+1
⌃ [e99dba38] Qt6Wayland_jll v6.7.1+1
  [f50d1b31] Rmath_jll v0.5.1+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
⌃ [a2964d1f] Wayland_jll v1.21.0+2
  [2381bf8a] Wayland_protocols_jll v1.36.0+0
⌅ [02c8fc9c] XML2_jll v2.13.6+1
  [ffd25f8a] XZ_jll v5.8.1+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.12+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.7+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.1+0
  [a51aa0fd] Xorg_libXi_jll v1.8.3+0
  [d1454406] Xorg_libXinerama_jll v1.1.6+0
  [ec84b674] Xorg_libXrandr_jll v1.5.5+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.1.3+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.4+0
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.44.0+0
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.9+0
  [214eeab7] fzf_jll v0.61.1+0
  [1a1c6b14] gperf_jll v3.1.1+1
  [9a68df92] isoband_jll v0.2.3+0
  [477f73a3] libaec_jll v1.1.3+0
  [a4ae2306] libaom_jll v3.11.0+0
  [0ac62f75] libass_jll v0.15.2+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [2db6ffa8] libevdev_jll v1.11.0+0
  [f638f0a6] libfdk_aac_jll v2.0.3+0
  [36db933b] libinput_jll v1.18.0+0
  [b53b4c65] libpng_jll v1.6.47+0
  [075b6546] libsixel_jll v1.10.5+0
  [f27f6e37] libvorbis_jll v1.3.7+2
  [c5f90fcd] libwebp_jll v1.5.0+0
  [009596ad] mtdev_jll v1.1.6+0
  [1317d2d5] oneTBB_jll v2022.0.0+0
⌅ [1270edf5] x264_jll v2021.5.5+0
⌅ [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+2
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [8bf52ea8] CRC32c
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.1+4
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.52.0+1
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`

You can also download the manifest file and the project file.