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.
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:
using ModelingToolkit
# Define some variables
@parameters t σ ρ β
@variables x(t) y(t) z(t)
@derivatives D'~tThen we build the system:
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 DiffEqSystem via:
de = DiffEqSystem(eqs,t,[x,y,z],[σ,ρ,β])
de = DiffEqSystem(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:
generate_function(de)
## 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:
f = ODEFunction(de)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.
@variables x y z
@parameters σ ρ β
# Define a nonlinear system
eqs = [0 ~ σ*(y-x),
0 ~ x*(ρ-z)-y,
0 ~ x*y - β*z]
ns = NonlinearSystem(eqs)
nlsys_func = generate_function(ns)which generates:
:((##364, u, p)->begin
let (x, z, y, ρ, σ, 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:
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))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.
The most fundamental part of the IR is the Variable. The Variable is the
context-aware single variable of the IR. Its fields are described as follows:
name: the name of theVariable. 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 theVariablein the sense of equality.known: the main denotation of context, storing whether or not the value of the variable is known.dependents: the vector of variables on which the current variable is dependent. For example,u(t,x)has dependents[t,x]. Derivatives thus require this information in order to simplify down.
Constant is a simple wrapper type to store numerical Julia constants.
Operations are the basic composition of variables and puts together the pieces with a function.
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.
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.
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:
eqsis the first argument which is an array ofOperationwhich 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 theeqscan 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.
Transformation functions send IR objects to like IR objects. These utilize the
contextual information in a given Operation/System to build another
Operation/System.
A function is registered into the operation system via:
@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.
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:
f(x,y,z) = x^2 + sin(x+y) - zautomatically 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:
# `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:
ModelingToolkit.derivative(::typeof(sin), args::NTuple{1,Any}, ::Val{1}) = cos(args[1])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:
@parameters t σ ρ β
@variables x(t) y(t) z(t)
@derivatives D'~tis syntactic sugar for:
t = Variable(:t; known = true)
x = Variable(:x, [t])
y = Variable(:y, [t])
z = Variable(:z, [t])
D = Differential(t)
σ = Variable(:σ; known = true)
ρ = Variable(:ρ; known = true)
β = Variable(:β; known = true)The system building functions can handle intermediate calculations. For example,
@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],[σ,ρ,β])
nlsys_func = generate_function(ns)expands to:
:((##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.