Parsing Expressions into Solvable Systems

Many times when creating DSLs or creating ModelingToolkit extensions to read new file formats, it can become imperative to parse expressions. In many cases, it can be easy to use Base.parse to take things to standard Julia expressions, but how can you take a Base.Expr and generate symbolic forms from that? For example, say we had the following system we wanted to solve:

ex = [:(y ~ x)
      :(y ~ -2x + 3 / z)
      :(z ~ 2)]

3-element Vector{Expr}:
 :(y ~ x)
 :(y ~ -2x + 3 / z)
 :(z ~ 2)

We can use the function parse_expr_to_symbolic from Symbolics.jl to generate the symbolic form of the expression:

using Symbolics
eqs = parse_expr_to_symbolic.(ex, (Main,))

\[ \begin{align} y =& x \\ y =& - 2 x + \frac{3}{z} \\ z =& 2 \end{align} \]

From there, we can use ModelingToolkit to transform the symbolic equations into a numerical nonlinear solve:

using ModelingToolkit, SymbolicIndexingInterface, NonlinearSolve
vars = union(ModelingToolkit.vars.(eqs)...)
@mtkbuild ns = NonlinearSystem(eqs, vars, [])

varmap = Dict(SymbolicIndexingInterface.getname.(vars) .=> vars)
prob = NonlinearProblem(ns, [varmap[:x] => 1.0, varmap[:y] => 1.0, varmap[:z] => 1.0])
sol = solve(prob, NewtonRaphson())

retcode: Success
u: 1-element Vector{Float64}:
 0.5