docs: add tutorial for callable parameters

Author: AayushSabharwal

docs/pages.jl

@@ -12,7 +12,8 @@ pages = [
         "tutorials/parameter_identifiability.md",
         "tutorials/bifurcation_diagram_computation.md",
         "tutorials/SampledData.md",
-        "tutorials/domain_connections.md"],
+        "tutorials/domain_connections.md",
+        "tutorials/callable_params.md"],
     "Examples" => Any[
         "Basic Examples" => Any["examples/higher_order.md",
             "examples/spring_mass.md",

docs/src/tutorials/callable_params.md

@@ -0,0 +1,90 @@
+# Callable parameters and interpolating data
+
+ModelingToolkit.jl allows creating parameters that represent functions to be called. This
+is especially useful for including interpolants and/or lookup tables inside ODEs. In this
+tutorial we will create an `ODESystem` which employs callable parameters to interpolate data
+inside an ODE and go over the various syntax options and their implications.
+
+## Callable parameter syntax
+
+The syntax for callable parameters declared via `@parameters` must be one of the following
+
+1. `(fn::FType)(..)`
+2. `fn(::argType1, ::argType2, ...)`
+
+In the first case, the parameter is callable with any number/combination of arguments, and
+has a type of `FType` (the callable must be a subtype of `FType`). In the second case,
+the parameter is callable with as many arguments as declared, and all must match the
+declared types.
+
+By default, the return type of the callable symbolic is inferred to be `Real`. To change
+this, a `::retType` annotation can be added at the end.
+
+To declare a function that returns an array of values, the same array syntax can be used
+as for normal variables:
+
+```julia
+@parameters (foo::FType)(..)[1:3]::retType
+@parameters foo(::argType1, ::argType2)[1:3]::retType
+```
+
+`retType` here is the `eltype` of the retured array.
+
+## Storage of callable parameters
+
+Callable parameters declared with the `::FType` syntax will be stored in a `Vector{FType}`.
+Thus, if `FType` is non-concrete, the buffer will also be non-concrete. This is sometimes
+necessary to allow the value of the callable to be switched out for a different type without
+rebuilding the model. Typically this syntax is preferable when `FType` is concrete or
+a small union.
+
+Callable parameters declared with the `::argType1, ...` syntax will be stored in a
+`Vector{FunctionWrappers.FunctionWrapper{retType, Tuple{argType1, ...}}}`. This suffers
+the small overhead of a `FunctionWrapper` and restricts the signature of the callable,
+symbolic, but allows storing the parameter in a type-stable manner and swapping it out.
+This is preferable when the values that the callable can take do not share a common
+subtype. For example, when a callable can represent the activation of a neural network
+and can be `tanh`, `sigmoid`, etc. which have a common ancestor of `Function`.
+
+If both `::FType` and `::argType`s are specified, `::FType` takes priority. For example,
+
+```julia
+@parameters (p::LinearInterpolation)(::Real)
+```
+
+`p` will be stored in a `Vector{LinearInterpolation}`. If `::LinearInterpolation` was not
+specified, it would be stored in a `Vector{FunctionWrapper{Real, Tuple{Real}}}`.
+
+## Example using interpolations
+
+```@example callable
+using ModelingToolkit
+using OrdinaryDiffEq
+using DataInterpolations
+using ModelingToolkit: t_nounits as t, D_nounits as D
+
+ts = collect(0.0:0.1:10.0)
+spline = LinearInterpolation(ts .^ 2, ts)
+Tspline = typeof(spline)
+@variables x(t)
+@parameters (interp::Tspline)(..)
+
+@mtkbuild sys = ODESystem(D(x) ~ interp(t), t)
+```
+
+The derivative of `x` is obtained via an interpolation from DataInterpolations.jl. Note
+the parameter syntax. The `(..)` marks the parameter as callable. `(interp::Tspline)`
+indicates that the parameter is of type `Tspline`.
+
+```@example callable
+prob = ODEProblem(sys, [x => 0.0], (0.0, 1.0), [interp => spline])
+solve(prob)
+```
+
+Note that the the following will not work:
+
+```julia
+ODEProblem(sys; [x => 0.0], (0.0, 1.0), [interp => LinearInterpolation(0.0:0.1:1.0, 0.0:0.1:1.0)])
+```
+
+Since the type of the spline doesn't match.