The AbstractSystem Interface

Overview

The AbstractSystem interface is the core of the system level of ModelingToolkit.jl. It establishes a common set of functionality that is used between systems from ODEs and chemical reactions, allowing users to have a common framework for model manipulation and compilation.

Composition and Accessor Functions

Each AbstractSystem has lists of variables in context, such as distinguishing parameters vs states. In addition, an AbstractSystem also can hold other AbstractSystem types. Direct accessing of the values, such as sys.states, gives the immediate list, while the accessor functions states(sys) gives the total set, which includes that of all systems held inside.

The values which are common to all AbstractSystems are:

  • sys.eqs or equations(sys): The equations that define the system.
  • sys.states or states(sys): The set of states in the system.
  • sys.parameters or parameters(sys): The parameters of the system.
  • sys.systems: The subsystems of the system.

Transformations

Transformations are functions which send a valid AbstractSystem definition to another AbstractSystem. These are passes, like optimizations (e.g., Block-Lower Triangle transformations), or changes to the representation, which allow for alternative numerical methods to be utilized on the model (e.g., DAE index reduction).

Function Calculation and Generation

The calculation and generation functions allow for calculating additional quantities to enhance the numerical methods applied to the resulting system. The calculations, like calculate_jacobian, generate ModelingToolkit IR for the Jacobian of the system, while the generations, like generate_jacobian, generate compiled output for the numerical solvers by applying build_function to the generated code. Additionally, many systems have function-type outputs, which cobble together the generation functionality for a system, for example, ODEFunction can be used to generate a DifferentialEquations-based ODEFunction with compiled version of the ODE itself, the Jacobian, the mass matrix, etc.

Below are the possible calculation and generation functions:

ModelingToolkit.calculate_tgradFunction
calculate_tgrad(sys::AbstractSystem)

Calculate the time gradient of a system.

Returns a vector of Expression instances. The result from the first call will be cached in the system object.

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ModelingToolkit.calculate_hessianFunction
calculate_hessian(sys::AbstractSystem)

Calculate the hessian matrix of a scalar system.

Returns a matrix of Expression instances. The result from the first call will be cached in the system object.

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ModelingToolkit.generate_tgradFunction
generate_tgrad(sys::AbstractSystem, dvs = states(sys), ps = parameters(sys), expression = Val{true}; kwargs...)

Generates a function for the time gradient of a system. Extra arguments control the arguments to the internal build_function call.

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ModelingToolkit.generate_gradientFunction
generate_gradient(sys::AbstractSystem, dvs = states(sys), ps = parameters(sys), expression = Val{true}; kwargs...)

Generates a function for the gradient of a system. Extra arguments control the arguments to the internal build_function call.

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ModelingToolkit.generate_jacobianFunction
generate_jacobian(sys::AbstractSystem, dvs = states(sys), ps = parameters(sys), expression = Val{true}; sparse = false, kwargs...)

Generates a function for the jacobian matrix matrix of a system. Extra arguments control the arguments to the internal build_function call.

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ModelingToolkit.generate_factorized_WFunction
generate_factorized_W(sys::AbstractSystem, dvs = states(sys), ps = parameters(sys), expression = Val{true}; sparse = false, kwargs...)

Generates a function for the factorized W-matrix matrix of a system. Extra arguments control the arguments to the internal build_function call.

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ModelingToolkit.generate_hessianFunction
generate_hessian(sys::AbstractSystem, dvs = states(sys), ps = parameters(sys), expression = Val{true}; sparse = false, kwargs...)

Generates a function for the hessian matrix matrix of a system. Extra arguments control the arguments to the internal build_function call.

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Additionally, jacobian_sparsity(sys) and hessian_sparsity(sys) exist on the appropriate systems for fast generation of the sparsity patterns via an abstract interpretation without requiring differentiation.

Problem Constructors

At the end, the system types have DEProblem constructors, like ODEProblem, which allow for directly generating the problem types required for numerical methods. The first argument is always the AbstractSystem, and the proceeding arguments match the argument order of their original constructors. Whenever an array would normally be provided, such as u0 the initial condition of an ODEProblem, it is instead replaced with a variable map, i.e., an array of pairs var=>value, which allows the user to designate the values without having to know the order that ModelingToolkit is internally using.