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 representing ODEs, PDEs, SDEs and more, allowing users to have a common framework for model manipulation and compilation.

Subtypes

There are three immediate subtypes of AbstractSystem, classified by how many independent variables each type has:

• AbstractTimeIndependentSystem: has no independent variable (eg: NonlinearSystem)
• AbstractTimeDependentSystem: has a single independent variable (eg: ODESystem)
• AbstractMultivariateSystem: may have multiple independent variables (eg: PDESystem)

Constructors and Naming

The AbstractSystem interface has a consistent method for constructing systems. Generally it follows the order of:

1. Equations
2. Independent Variables
3. Dependent Variables (or States)
4. Parameters

All other pieces are handled via keyword arguments. AbstractSystems share the same keyword arguments, which are:

• system: This is used for specifying subsystems for hierarchical modeling with reusable components. For more information, see the components page
• Defaults: Keyword arguments like defaults are used for specifying default values which are used. If a value is not given at the SciMLProblem construction time, its numerical value will be the default.

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:

• equations(sys): All equations that define the system and its subsystems.
• states(sys): All the states in the system and its subsystems.
• parameters(sys): All parameters of the system and its subsystems.
• nameof(sys): The name of the current-level system.
• get_eqs(sys): Equations that define the current-level system.
• get_states(sys): States that are in the current-level system.
• get_ps(sys): Parameters that are in the current-level system.
• get_systems(sys): Subsystems of the current-level system.

Optionally, a system could have:

• observed(sys): All observed equations of the system and its subsystems.
• get_observed(sys): Observed equations of the current-level system.
• get_continuous_events(sys): SymbolicContinuousCallbacks of the current-level system.
• get_defaults(sys): A Dict that maps variables into their default values.
• independent_variables(sys): The independent variables of a system.
• get_noiseeqs(sys): Noise equations of the current-level system.

Note that if you know a system is an AbstractTimeDependentSystem you could use get_iv to get the unique independent variable directly, rather than using independenent_variables(sys)[1], which is clunky and may cause problems if sys is an AbstractMultivariateSystem because there may be more than one independent variable. AbstractTimeIndependentSystems do not have a method get_iv, and independent_variables(sys) will return a size-zero result for such. For an AbstractMultivariateSystem, get_ivs is equivalent.

A system could also have caches:

• get_jac(sys): The Jacobian of a system.
• get_tgrad(sys): The gradient with respect to time of a 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).

Analyses

Analyses are functions on a system which return information about the corresponding properties, like whether its parameters are structurally identifiable, or whether it's linear.

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:

calculate_tgrad(sys::AbstractTimeDependentSystem)

calculate_gradient(sys::AbstractSystem)

calculate_jacobian(sys::AbstractSystem)

calculate_factorized_W(sys::AbstractSystem)

calculate_hessian(sys::AbstractSystem)

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

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

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

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

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

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.

For the value maps, the parameters are allowed to be functions of each other, and value maps of states can be functions of the parameters, i.e. you can do:

u0 = [
  lorenz1.x => 2.0
  lorenz2.x => lorenz1.x * lorenz1.p
]

Default Value Handling

The AbstractSystem types allow for specifying default values, for example defaults inside of them. At problem construction time, these values are merged into the value maps, where for any repeats the value maps override the default. In addition, defaults of a higher level in the system override the defaults of a lower level in the system.