ModelingToolkit.jl

ModelingToolkit.jl is an intermediate representation (IR) of computational graphs for scientific computing problems. Its purpose is to be a common target for modeling DSLs in order to allow for a common platform for model inspection and transformation. It uses a tagged variable IR in order to allow specification of complex models and allow for transformations of models. It has ways to plug into its function registration and derivative system so that way it can interact nicely with user-defined routines. Together, this is an abstract form of a scientific model that is easy for humans to generate but also easy for programs to manipulate.

Introduction by Examples

Example: ODE

Let's build an ODE. First we define some variables. In a differential equation system, we need to differentiate between our (dependent) variables and parameters. Therefore we label them as follows:

```julia using ModelingToolkit

Define some variables

@parameters t σ ρ β @variables x(t) y(t) z(t) @derivatives D'~t ```

Then we build the system:

julia eqs = [D(x) ~ σ*(y-x), D(y) ~ x*(ρ-z)-y, D(z) ~ x*y - β*z]

Each operation builds an Operation type, and thus eqs is an array of Operation and Variables. This holds a tree of the full system that can be analyzed by other programs. We can turn this into a ODESystem via:

julia de = ODESystem(eqs)

where we tell it the variable types and ordering in the first version, or let it automatically determine the variable types in the second version. This can then generate the function. For example, we can see the generated code via:

```julia generate_function(de, [x,y,z], [σ,ρ,β])

Which returns:

:((##363, u, p, t)->begin let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) ##363[1] = σ * (y - x) ##363[2] = x * (ρ - z) - y ##363[3] = x * y - β * z end end) ```

and get the generated function via:

julia f = ODEFunction(de, [x,y,z], [σ,ρ,β])

Example: Nonlinear System

We can also build nonlinear systems. Let's say we wanted to solve for the steady state of the previous ODE. This is the nonlinear system defined by where the derivatives are zero. We use (unknown) variables for our nonlinear system.

```julia @variables x y z @parameters σ ρ β

Define a nonlinear system

eqs = [0 ~ σ(y-x), 0 ~ x(ρ-z)-y, 0 ~ xy - βz] ns = NonlinearSystem(eqs, [x,y,z]) nlsysfunc = generatefunction(ns, [x,y,z], [σ,ρ,β]) ```

which generates:

julia :((##364, u, p)->begin let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) ##364[1] = σ * (y - x) ##364[2] = x * (ρ - z) - y ##364[3] = x * y - β * z end end)

We can use this to build a nonlinear function for use with NLsolve.jl:

```julia f = @eval eval(nlsys_func)

Make a closure over the parameters for for NLsolve.jl

f2 = (du,u) -> f(du,u,(10.0,26.0,2.33)) ```

Example: Arrays of variables

Sometimes it is convenient to define arrays of variables to model things like x₁,…,x₃. The @variables and @parameters macros support this with the following syntax:

```julia julia> @variables x[1:3]; julia> x 3-element Array{Operation,1}: x₁() x₂() x₃()

support for arbitrary ranges and tensors

julia> @variables y[2:3,1:5:6]; julia> y 2×2 Array{Operation,2}: y₂̒₁() y₂̒₆() y₃̒₁() y₃̒₆()

also works for dependent variables

julia> @parameters t; @variables z1:3; julia> z 3-element Array{Operation,1}: z₁(t()) z₂(t()) z₃(t()) ```

Core Principles

The core idea behind ModelingToolkit.jl is that mathematical equations require context, and thus any symbolic manipulations and full model specifications requires the ability to handle such context. When writing DSLs, this fact comes to light very quickly. Every DSL seems to lower to some intermediate representation from which the final result is computed, but this process means there's a lot of repeated ideas for every DSL that creates scientific computing objects like differential equations and nonlinear systems. By having a single common contexualized IR, this gives DSLs a target to write to so that way lower-level details like computation of system Jacobians can be disconnected from the DSL and its syntax, allowing for code-reuse between modeling packages and languages.

In this section we define the core pieces of the IR and what they mean.

Variables

The most fundamental part of the IR is the Variable. In order to mirror the intention of solving for variables and representing function-like parameters, we treat each instance of Variable as a function which is called on its arguments using the natural syntax. Rather than having additional mechanisms for handling constant variables and parameters, we simply represent them as constant functions.

The Variable is the context-aware single variable of the IR. Its fields are described as follows:

• name: the name of the Variable. Note that this is not necessarily the same as the name of the Julia variable. But this symbol itself is considered the core identifier of the Variable in the sense of equality.
• known: the main denotation of context, storing whether or not the value of the variable is known.

For example, the following code defines an independent variable t, a parameter α, a function parameter σ, a variable x which depends on t, a variable y with no dependents, a variable z which depends on t, α, and x(t) and a parameters β₁ and β₂.

``julia t = Variable(:t; known = true)() # independent variables are treated as known α = Variable(:α; known = true)() # parameters are known σ = Variable(:σ; known = true) # left uncalled, since it is used as a function w = Variable(:w; known = false) # unknown, left uncalled x = Variable(:x; known = false)(t) # unknown, depends ont` y = Variable(:y; known = false)() # unknown, no dependents z = Variable(:z; known = false)(t, α, x) # unknown, multiple arguments β₁ = Variable(:β, 1; known = true)() # with index 1 β₂ = Variable(:β, 2; known = true)() # with index 2

expr = β₁ * x + y^α + σ(3) * (z - t) - β₂ * w(t - 1) ```

We can rewrite this more concisely using macros. Note the difference between including and excluding empty parentheses. When in call format, variables are aliased to the given call, allowing implicit use of dependents for convenience.

```julia @parameters t α σ(..) β[1:2] @variables w(..) x(t) y() z(t, α, x)

expr = β₁* x + y^α + σ(3) * (z - t) - β₂ * w(t - 1) ```

Note that @parameters and @variables implicitly add () to values that are not given a call. The former specifies the values as known, while the latter specifies it as unknown. (..) signifies that the value should be left uncalled.

Constants

Constant is a simple wrapper type to store numerical Julia constants.

Operations

Operations are the basic composition of variables and puts together the pieces with a function.

Equations

Equations are stored using the Equation datatype. Given expressions for the left-hand and right-hand sides, an equation is constructed as Equation(lhs, rhs), or equivalently lhs ~ rhs.

Differentials

A Differential denotes the derivative with respect to a given variable. It can be expanded via expand_derivatives, which symbolically differentiates expressions recursively and cancels out appropriate constant variables.

Systems

A system is a collection of operations with expanded context. While different systems can have different constructors and interpretations, the general structure is as follows:

• eqs is the first argument which is an array of Operation which describe the system of equations.
• Name to subtype mappings: these describe how variable subtypes are mapped to the contexts of the system. For example, for a differential equation, the variable corresponds to given subtypes and then the eqs can be analyzed knowing what the state variables are.
• Variable names which do not fall into one of the system's core subtypes are treated as intermediates which can be used for holding subcalculations and other pieces like that.

Transformations

Transformation functions send IR objects to like IR objects. These utilize the contextual information in a given Operation/System to build another Operation/System.

Details

Function Registration

A function is registered into the operation system via:

julia @register f(x) @register g(x,y)

etc. where each macro call registers the function with the given signature. This will cause operations to stop recursing at this function, building Operation(g,args) nodes into the graph instead of tracing calls of g itself into Operations.

Adding Derivatives

There is a large amount of derivatives pre-defined by DiffRules.jl. Note that Expression types are defined as <:Real, and thus any functions which allow the use of real numbers can automatically be traced by the derivative mechanism. Thus for example:

julia f(x,y,z) = x^2 + sin(x+y) - z

automatically has the derivatives defined via the tracing mechanism. It will do this by directly building the operation the internals of your function and differentiating that.

However, in many cases you may want to define your own derivatives so that way automatic Jacobian etc. calculations can utilize this information. This can allow for more succinct versions of the derivatives to be calculated in order to better scale to larger systems. You can define derivatives for your own function via the dispatch:

```julia

N arguments are accepted by the relevant method of my_function

ModelingToolkit.derivative(::typeof(my_function), args::NTuple{N,Any}, ::Val{i}) ```

where i means that it's the derivative of the ith argument. args is the array of arguments, so for example if your function is f(x,t) then args = [x,t]. You should return an Operation for the derivative of your function.

For example, sin(t)'s derivative (by t) is given by the following:

julia ModelingToolkit.derivative(::typeof(sin), args::NTuple{1,Any}, ::Val{1}) = cos(args[1])

Macro-free Usage

Given the insistence on being programming friendly, all of the functionality is accessible via a function-based interface. This means that all macros are syntactic sugar in some form. For example, the variable construction:

julia @parameters t σ ρ β[1:3] @variables x(t) y(t) z(t) @derivatives D'~t

is syntactic sugar for:

julia t = Variable(:t; known = true)() σ = Variable(:σ; known = true)() ρ = Variable(:ρ; known = true)() β = [Variable(:β, i; known = true)() for i in 1:3] x = Variable(:x)(t) y = Variable(:y)(t) z = Variable(:z)(t) D = Differential(t)

Intermediate Calculations

The system building functions can handle intermediate calculations. For example,

julia @variables x y z @parameters σ ρ β a = y - x eqs = [0 ~ σ*a, 0 ~ x*(ρ-z)-y, 0 ~ x*y - β*z] ns = NonlinearSystem(eqs, [x,y,z]) nlsys_func = generate_function(ns, [x,y,z], [σ,ρ,β])

expands to:

julia :((##365, u, p)->begin let (x, y, z, σ, ρ, β) = (u[1], u[2], u[3], p[1], p[2], p[3]) ##365[1] = σ * (y - x) ##365[2] = x * (ρ - z) - y ##365[3] = x * y - β * z end end)

In addition, the Jacobian calculations take into account intermediate variables to appropriately handle them.