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NEWS.md

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- `connect` should not be overloaded by users anymore. `[connect = Flow]`
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informs ModelingToolkit that particular variable in a connector ought to sum
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to zero, and by default, variables are equal in a connection. Please check out
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[acausal components tutorial](https://mtk.sciml.ai/dev/tutorials/acausal_components/)
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[acausal components tutorial](https://docs.sciml.ai/ModelingToolkit/stable/tutorials/acausal_components/)
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for examples.

README.md

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to the model to make it easier for numerical solvers to handle.
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For information on using the package,
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[see the stable documentation](https://mtk.sciml.ai/stable/). Use the
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[in-development documentation](https://mtk.sciml.ai/dev/) for the version of
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[see the stable documentation](https://docs.sciml.ai/ModelingToolkit/stable/). Use the
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[in-development documentation](https://docs.sciml.ai/ModelingToolkit/dev/) for the version of
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the documentation which contains the unreleased features.
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## Standard Library
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For a standard library of ModelingToolkit components and blocks, check out the
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[ModelingToolkitStandardLibrary](https://github.com/SciML/ModelingToolkitStandardLibrary.jl)
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[ModelingToolkitStandardLibrary](https://docs.sciml.ai/ModelingToolkitStandardLibrary/stable/)
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## High-Level Examples
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docs/make.jl

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format = Documenter.HTML(; analytics = "UA-90474609-3",
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assets = ["assets/favicon.ico"],
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mathengine,
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canonical = "https://mtk.sciml.ai/stable/",
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canonical = "https://docs.sciml.ai/ModelingToolkit/stable/",
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prettyurls = (get(ENV, "CI", nothing) == "true")),
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pages = pages)
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docs/src/basics/ContextualVariables.md

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helping the system transformation machinery do complex manipulations and
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automatic detection. The standard variable definition in ModelingToolkit.jl is
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the `@variable` which is defined by
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[Symbolics.jl](https://github.com/JuliaSymbolics/Symbolics.jl). For example:
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[Symbolics.jl](https://docs.sciml.ai/Symbolics/stable/). For example:
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```julia
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@variables x y(x)

docs/src/basics/Events.md

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system state or parameters to be changed when certain conditions are satisfied,
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or can be used to detect discontinuities. These events are ultimately converted
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into DifferentialEquations.jl [`ContinuousCallback`s or
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`DiscreteCallback`s](https://docs.sciml.ai/stable/modules/DiffEqDocs/features/callback_functions/),
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`DiscreteCallback`s](https://docs.sciml.ai/DiffEqDocs/stable/features/callback_functions/),
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or into more specialized callback types from the
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[DiffEqCallbacks.jl](https://docs.sciml.ai/stable/modules/DiffEqDocs/features/callback_library/)
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[DiffEqCallbacks.jl](https://docs.sciml.ai/DiffEqDocs/stable/features/callback_library/)
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library.
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[`ODESystem`](@ref)s and [`SDESystem`](@ref)s accept keyword arguments
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zero, with root finding used to determine the time at which a zero crossing
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occurred. Discrete events are applied when a condition tested after each
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timestep evaluates to true. See the [DifferentialEquations
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docs](https://docs.sciml.ai/stable/modules/DiffEqDocs/features/callback_functions/)
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docs](https://docs.sciml.ai/DiffEqDocs/stable/features/callback_functions/)
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for more detail.
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Events involve both a *condition* function (for the zero crossing or truth
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### Example: Bouncing ball
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In the documentation for
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[DifferentialEquations](https://docs.sciml.ai/stable/modules/DiffEqDocs/features/callback_functions/#Example-1:-Bouncing-Ball),
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[DifferentialEquations](https://docs.sciml.ai/DiffEqDocs/stable/features/callback_functions/#Example-1:-Bouncing-Ball),
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we have an example where a bouncing ball is simulated using callbacks which have
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an `affect!` on the state. We can model the same system using ModelingToolkit
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like this
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passed to `affect!` as the `ctx` argument.
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`affect!` receives a [DifferentialEquations.jl
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integrator](https://docs.sciml.ai/stable/modules/DiffEqDocs/basics/integrator/)
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integrator](https://docs.sciml.ai/DiffEqDocs/stable/basics/integrator/)
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as its first argument, which can then be used to access states and parameters
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that are provided in the `u` and `p` arguments (implemented as `NamedTuple`s).
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The integrator can also be manipulated more generally to control solution
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behavior, see the [integrator
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interface](https://docs.sciml.ai/stable/modules/DiffEqDocs/basics/integrator/)
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interface](https://docs.sciml.ai/DiffEqDocs/stable/basics/integrator/)
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documentation. In affect functions we have that
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```julia
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function affect!(integ, u, p, ctx)

docs/src/basics/Linearization.md

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## Tools for linear analysis
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[ModelingToolkitStandardLibrary](http://mtkstdlib.sciml.ai/dev/API/linear_analysis/) contains a set of [tools for more advanced linear analysis](http://mtkstdlib.sciml.ai/dev/API/linear_analysis/). These can be used to make it easier to work with and analyze causal models, such as control and signal-processing systems.
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[ModelingToolkitStandardLibrary](https://docs.sciml.ai/ModelingToolkitStandardLibrary/stable/) contains a set of [tools for more advanced linear analysis](https://docs.sciml.ai/ModelingToolkitStandardLibrary/stable/API/linear_analysis/). These can be used to make it easier to work with and analyze causal models, such as control and signal-processing systems.
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```@index
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Pages = ["Linearization.md"]

docs/src/comparison.md

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Julia as an order of magnitude or more faster in many cases due to its JIT
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compilation.
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- Simulink uses the MATLAB differential equation solvers while ModelingToolkit.jl
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uses [DifferentialEquations.jl](https://diffeq.sciml.ai/dev/). For a systematic
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uses [DifferentialEquations.jl](https://docs.sciml.ai/DiffEqDocs/stable/). For a systematic
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comparison between the solvers, consult
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[open benchmarks](https://benchmarks.sciml.ai/html/MultiLanguage/wrapper_packages.html)
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[open benchmarks](https://docs.sciml.ai/SciMLBenchmarksOutput/stable/)
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which demonstrate two orders of magnitude performance advantage for the native
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Julia solvers across many benchmark problems.
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- Simulink comes with a Graphical User Interface (GUI), ModelingToolkit.jl
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interactively in the REPL.
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- CASADI includes limited support for Computer Algebra System (CAS) functionality,
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while ModelingToolkit.jl is built on the full
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[Symbolics.jl](https://github.com/JuliaSymbolics/Symbolics.jl) CAS.
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[Symbolics.jl](https://docs.sciml.ai/Symbolics/stable/) CAS.
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- CASADI supports DAE and ODE problems via SUNDIALS IDAS and CVODES. ModelingToolkit.jl
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supports DAE and ODE problems via [DifferentialEquations.jl](https://diffeq.sciml.ai/dev/),
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supports DAE and ODE problems via [DifferentialEquations.jl](https://docs.sciml.ai/DiffEqDocs/stable/),
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of which Sundials.jl is <1% of the total available solvers and is outperformed
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by the native Julia solvers on the vast majority of the benchmark equations.
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In addition, the DifferentialEquations.jl interface is confederated, meaning
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## Comparison Against Modia.jl
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- Modia.jl uses Julia's expression objects for representing its equations.
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ModelingToolkit.jl uses [Symbolics.jl](https://symbolics.juliasymbolics.org/dev/),
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ModelingToolkit.jl uses [Symbolics.jl](https://docs.sciml.ai/Symbolics/stable/),
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and thus the Julia expressions follow Julia semantics and can be manipulated
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using a computer algebra system (CAS).
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- Modia's compilation pipeline is similar to the
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`structural_simplify` is an adaptation of the Modia.jl-improved alias elimination
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and tearing algorithms.
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- Both Modia and ModelingToolkit generate `DAEProblem` and `ODEProblem` forms for
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solving with [DifferentialEquations.jl](https://diffeq.sciml.ai/dev/).
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solving with [DifferentialEquations.jl](https://docs.sciml.ai/DiffEqDocs/stable/).
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- ModelingToolkit.jl integrates with its host language Julia, so Julia code
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can be automatically converted into ModelingToolkit expressions. Users of
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Modia must explicitly create Modia expressions.
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- Causal.jl is a causal modeling environment, whereas ModelingToolkit.jl is an
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acausal modeling environment. For an overview of the differences, consult
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academic reviews such as [this one](https://arxiv.org/abs/1909.00484).
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- Both ModelingToolkit.jl and Causal.jl use [DifferentialEquations.jl](https://diffeq.sciml.ai/stable/)
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- Both ModelingToolkit.jl and Causal.jl use [DifferentialEquations.jl](https://docs.sciml.ai/DiffEqDocs/stable/)
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as the backend solver library.
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- Causal.jl lets one add arbitrary equation systems to a given node, and allow
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the output to effect the next node. This means an SDE may drive an ODE. These

docs/src/index.md

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For information on how to use the Symbolics.jl CAS system that ModelingToolkit.jl
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[Symbolics.jl documentation](https://docs.sciml.ai/Symbolics/stable/)
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### Equation Types
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- Nonlinear systems
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- Optimization problems
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- Continuous-Time Markov Chains
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- Chemical Reactions (via [Catalyst.jl](https://github.com/SciML/Catalyst.jl))
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- Chemical Reactions (via [Catalyst.jl](https://docs.sciml.ai/Catalyst/stable/))
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- Nonlinear Optimal Control
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## Standard Library
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For quick development, ModelingToolkit.jl includes
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[ModelingToolkitStandardLibrary.jl](https://github.com/SciML/ModelingToolkitStandardLibrary.jl),
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[ModelingToolkitStandardLibrary.jl](https://docs.sciml.ai/ModelingToolkitStandardLibrary/stable/),
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a standard library of prebuilt components for the ModelingToolkit ecosystem.
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## Model Import Formats
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- [CellMLToolkit.jl](https://github.com/SciML/CellMLToolkit.jl): Import [CellML](https://www.cellml.org/) models into ModelingToolkit
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- [CellMLToolkit.jl](https://docs.sciml.ai/CellMLToolkit/stable/): Import [CellML](https://www.cellml.org/) models into ModelingToolkit
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- Repository of more than a thousand pre-made models
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- Focus on biomedical models in areas such as: Calcium Dynamics,
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Cardiovascular Circulation, Cell Cycle, Cell Migration, Circadian Rhythms,
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Electrophysiology, Endocrine, Excitation-Contraction Coupling, Gene Regulation,
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Hepatology, Immunology, Ion Transport, Mechanical Constitutive Laws,
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Metabolism, Myofilament Mechanics, Neurobiology, pH Regulation, PKPD,
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Protein Modules, Signal Transduction, and Synthetic Biology.
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- [SBMLToolkit.jl](https://github.com/SciML/SBMLToolkit.jl): Import [SBML](http://sbml.org/) models into ModelingToolkit
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- [SBMLToolkit.jl](https://docs.sciml.ai/SBMLToolkit/stable/): Import [SBML](http://sbml.org/) models into ModelingToolkit
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- Uses the robust libsbml library for parsing and transforming the SBML
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- [ReactionNetworkImporters.jl](https://github.com/SciML/ReactionNetworkImporters.jl): Import various models into ModelingToolkit
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- [ReactionNetworkImporters.jl](https://docs.sciml.ai/ReactionNetworkImporters/stable/): Import various models into ModelingToolkit
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- Supports the BioNetGen `.net` file
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- Supports importing networks specified by stoichiometric matrices
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ecosystem, there is a large set of libraries adding features to this system.
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Below is an incomplete list of extension libraries one may want to be aware of:
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- [Catalyst.jl](https://github.com/SciML/Catalyst.jl): Symbolic representations
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- [Catalyst.jl](https://docs.sciml.ai/Catalyst/stable/): Symbolic representations
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of chemical reactions
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- Symbolically build and represent large systems of chemical reactions
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- Generate code for ODEs, SDEs, continuous-time Markov Chains, and more
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- Simulate the models using the SciML ecosystem with O(1) Gillespie methods
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- [DataDrivenDiffEq.jl](https://github.com/SciML/DataDrivenDiffEq.jl): Automatic
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- [DataDrivenDiffEq.jl](https://docs.sciml.ai/DataDrivenDiffEq/stable/): Automatic
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identification of equations from data
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- Automated construction of ODEs and DAEs from data
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- Representations of Koopman operators and Dynamic Mode Decomposition (DMD)
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- [MomentClosure.jl](https://github.com/augustinas1/MomentClosure.jl): Automatic
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- [MomentClosure.jl](https://docs.sciml.ai/MomentClosure/dev/): Automatic
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transformation of ReactionSystems into deterministic systems
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- Generates ODESystems for the moment closures
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- Allows for geometrically-distributed random reaction rates
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- [ReactionMechanismSimulator.jl](https://github.com/ReactionMechanismGenerator/ReactionMechanismSimulator.jl):
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- [ReactionMechanismSimulator.jl](https://docs.sciml.ai/ReactionMechanismSimulator/stable):
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Simulating and analyzing large chemical reaction mechanisms
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- Ideal gas and dilute liquid phases.
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- Constant T and P and constant V adiabatic ideal gas reactors.
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the solver libraries which are the numerical targets of the ModelingToolkit
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- [DifferentialEquations.jl](https://diffeq.sciml.ai/stable/)
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- [DifferentialEquations.jl](https://docs.sciml.ai/DiffEqDocs/stable/)
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- Multi-package interface of high performance numerical solvers for `ODESystem`,
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`SDESystem`, and `JumpSystem`
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- [NonlinearSolve.jl](https://github.com/JuliaComputing/NonlinearSolve.jl)
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- [NonlinearSolve.jl](https://docs.sciml.ai/NonlinearSolve/stable/)
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- High performance numerical solving of `NonlinearSystem`
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- [Optimization.jl](https://github.com/SciML/Optimization.jl)
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- [Optimization.jl](https://docs.sciml.ai/Optimization/stable/)
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- Multi-package interface for numerical solving `OptimizationSystem`
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- [NeuralPDE.jl](https://github.com/SciML/NeuralPDE.jl)
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- [NeuralPDE.jl](https://docs.sciml.ai/NeuralPDE/stable/)
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- Physics-Informed Neural Network (PINN) training on `PDESystem`
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- [MethodOfLines.jl](https://github.com/SciML/MethodOfLines.jl)
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- [MethodOfLines.jl](https://docs.sciml.ai/MethodOfLines/stable/)
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- Automated finite difference method (FDM) discretization of `PDESystem`
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## Contributing

docs/src/mtkitize_tutorials/modelingtoolkitize_index_reduction.md

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As a good DifferentialEquations.jl user, one would follow
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[the mass matrix DAE tutorial](https://diffeq.sciml.ai/stable/tutorials/dae_example/#Mass-Matrix-Differential-Algebraic-Equations-(DAEs))
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[the mass matrix DAE tutorial](https://docs.sciml.ai/DiffEqDocs/stable/tutorials/dae_example/#Mass-Matrix-Differential-Algebraic-Equations-(DAEs))
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to arrive at code for simulating the model:
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```@example indexred

docs/src/systems/PDESystem.md

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### NeuralPDE.jl: PhysicsInformedNN
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[NeuralPDE.jl](https://github.com/SciML/NeuralPDE.jl) defines the `PhysicsInformedNN`
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discretizer which uses a [DiffEqFlux.jl](https://github.com/SciML/DiffEqFlux.jl)
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[NeuralPDE.jl](https://docs.sciml.ai/NeuralPDE/stable/) defines the `PhysicsInformedNN`
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discretizer which uses a [DiffEqFlux.jl](https://docs.sciml.ai/DiffEqFlux/stable/)
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neural network to solve the differential equation.
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### MethodOfLines.jl: MOLFiniteDifference
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[MethodOfLines.jl](https://github.com/SciML/MethodOfLines.jl) defines the
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[MethodOfLines.jl](https://docs.sciml.ai/MethodOfLines/stable/) defines the
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`MOLFiniteDifference` discretizer which performs a finite difference discretization.
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Includes support for higher approximation order stencils and nonuniform grids.

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