Merge pull request #3249 from AayushSabharwal/as/hc-everywhere

feat: use `HomotopyContinuationProblem` in `NonlinearProblem` if possible

Author: ChrisRackauckas

ext/MTKHomotopyContinuationExt.jl

@@ -11,217 +11,6 @@ using ModelingToolkit: iscomplete, parameters, has_index_cache, get_index_cache,
 
 const MTK = ModelingToolkit
 
-function contains_variable(x, wrt)
-    any(y -> occursin(y, x), wrt)
-end
-
-"""
-Possible reasons why a term is not polynomial
-"""
-MTK.EnumX.@enumx NonPolynomialReason begin
-    NonIntegerExponent
-    ExponentContainsUnknowns
-    BaseNotPolynomial
-    UnrecognizedOperation
-end
-
-function display_reason(reason::NonPolynomialReason.T, sym)
-    if reason == NonPolynomialReason.NonIntegerExponent
-        pow = arguments(sym)[2]
-        "In $sym: Exponent $pow is not an integer"
-    elseif reason == NonPolynomialReason.ExponentContainsUnknowns
-        pow = arguments(sym)[2]
-        "In $sym: Exponent $pow contains unknowns of the system"
-    elseif reason == NonPolynomialReason.BaseNotPolynomial
-        base = arguments(sym)[1]
-        "In $sym: Base $base is not a polynomial in the unknowns"
-    elseif reason == NonPolynomialReason.UnrecognizedOperation
-        op = operation(sym)
-        """
-        In $sym: Operation $op is not recognized. Allowed polynomial operations are \
-        `*, /, +, -, ^`.
-        """
-    else
-        error("This should never happen. Please open an issue in ModelingToolkit.jl.")
-    end
-end
-
-mutable struct PolynomialData
-    non_polynomial_terms::Vector{BasicSymbolic}
-    reasons::Vector{NonPolynomialReason.T}
-    has_parametric_exponent::Bool
-end
-
-PolynomialData() = PolynomialData(BasicSymbolic[], NonPolynomialReason.T[], false)
-
-abstract type PolynomialTransformationError <: Exception end
-
-struct MultivarTerm <: PolynomialTransformationError
-    term::Any
-    vars::Any
-end
-
-function Base.showerror(io::IO, err::MultivarTerm)
-    println(io,
-        "Cannot convert system to polynomial: Found term $(err.term) which is a function of multiple unknowns $(err.vars).")
-end
-
-struct MultipleTermsOfSameVar <: PolynomialTransformationError
-    terms::Any
-    var::Any
-end
-
-function Base.showerror(io::IO, err::MultipleTermsOfSameVar)
-    println(io,
-        "Cannot convert system to polynomial: Found multiple non-polynomial terms $(err.terms) involving the same unknown $(err.var).")
-end
-
-struct SymbolicSolveFailure <: PolynomialTransformationError
-    term::Any
-    var::Any
-end
-
-function Base.showerror(io::IO, err::SymbolicSolveFailure)
-    println(io,
-        "Cannot convert system to polynomial: Unable to symbolically solve $(err.term) for $(err.var).")
-end
-
-struct NemoNotLoaded <: PolynomialTransformationError end
-
-function Base.showerror(io::IO, err::NemoNotLoaded)
-    println(io,
-        "ModelingToolkit may be able to solve this system as a polynomial system if `Nemo` is loaded. Run `import Nemo` and try again.")
-end
-
-struct VariablesAsPolyAndNonPoly <: PolynomialTransformationError
-    vars::Any
-end
-
-function Base.showerror(io::IO, err::VariablesAsPolyAndNonPoly)
-    println(io,
-        "Cannot convert convert system to polynomial: Variables $(err.vars) occur in both polynomial and non-polynomial terms in the system.")
-end
-
-struct NotPolynomialError <: Exception
-    transformation_err::Union{PolynomialTransformationError, Nothing}
-    eq::Vector{Equation}
-    data::Vector{PolynomialData}
-end
-
-function Base.showerror(io::IO, err::NotPolynomialError)
-    if err.transformation_err !== nothing
-        Base.showerror(io, err.transformation_err)
-    end
-    for (eq, data) in zip(err.eq, err.data)
-        if isempty(data.non_polynomial_terms)
-            continue
-        end
-        println(io,
-            "Equation $(eq) is not a polynomial in the unknowns for the following reasons:")
-        for (term, reason) in zip(data.non_polynomial_terms, data.reasons)
-            println(io, display_reason(reason, term))
-        end
-    end
-end
-
-function is_polynomial!(data, y, wrt)
-    process_polynomial!(data, y, wrt)
-    isempty(data.reasons)
-end
-
-"""
-$(TYPEDSIGNATURES)
-
-Return information about the polynmial `x` with respect to variables in `wrt`,
-writing said information to `data`.
-"""
-function process_polynomial!(data::PolynomialData, x, wrt)
-    x = unwrap(x)
-    symbolic_type(x) == NotSymbolic() && return true
-    iscall(x) || return true
-    contains_variable(x, wrt) || return true
-    any(isequal(x), wrt) && return true
-
-    if operation(x) in (*, +, -, /)
-        # `map` because `all` will early exit, but we want to search
-        # through everything to get all the non-polynomial terms
-        return all(map(y -> is_polynomial!(data, y, wrt), arguments(x)))
-    end
-    if operation(x) == (^)
-        b, p = arguments(x)
-        is_pow_integer = symtype(p) <: Integer
-        if !is_pow_integer
-            push!(data.non_polynomial_terms, x)
-            push!(data.reasons, NonPolynomialReason.NonIntegerExponent)
-        end
-        if symbolic_type(p) != NotSymbolic()
-            data.has_parametric_exponent = true
-        end
-
-        exponent_has_unknowns = contains_variable(p, wrt)
-        if exponent_has_unknowns
-            push!(data.non_polynomial_terms, x)
-            push!(data.reasons, NonPolynomialReason.ExponentContainsUnknowns)
-        end
-        base_polynomial = is_polynomial!(data, b, wrt)
-        return base_polynomial && !exponent_has_unknowns && is_pow_integer
-    end
-    push!(data.non_polynomial_terms, x)
-    push!(data.reasons, NonPolynomialReason.UnrecognizedOperation)
-    return false
-end
-
-"""
-$(TYPEDSIGNATURES)
-
-Given a `x`, a polynomial in variables in `wrt` which may contain rational functions,
-express `x` as a single rational function with polynomial `num` and denominator `den`.
-Return `(num, den)`.
-"""
-function handle_rational_polynomials(x, wrt)
-    x = unwrap(x)
-    symbolic_type(x) == NotSymbolic() && return x, 1
-    iscall(x) || return x, 1
-    contains_variable(x, wrt) || return x, 1
-    any(isequal(x), wrt) && return x, 1
-
-    # simplify_fractions cancels out some common factors
-    # and expands (a / b)^c to a^c / b^c, so we only need
-    # to handle these cases
-    x = simplify_fractions(x)
-    op = operation(x)
-    args = arguments(x)
-
-    if op == /
-        # numerator and denominator are trivial
-        num, den = args
-        # but also search for rational functions in numerator
-        n, d = handle_rational_polynomials(num, wrt)
-        num, den = n, den * d
-    elseif op == +
-        num = 0
-        den = 1
-
-        # we don't need to do common denominator
-        # because we don't care about cases where denominator
-        # is zero. The expression is zero when all the numerators
-        # are zero.
-        for arg in args
-            n, d = handle_rational_polynomials(arg, wrt)
-            num += n
-            den *= d
-        end
-    else
-        return x, 1
-    end
-    # if the denominator isn't a polynomial in `wrt`, better to not include it
-    # to reduce the size of the gcd polynomial
-    if !contains_variable(den, wrt)
-        return num / den, 1
-    end
-    return num, den
-end
-
 """
 $(TYPEDSIGNATURES)
 
@@ -289,12 +78,6 @@ end
 
 SymbolicIndexingInterface.parameter_values(s::MTKHomotopySystem) = s.p
 
-struct PolynomialTransformationData
-    new_var::BasicSymbolic
-    term::BasicSymbolic
-    inv_term::Vector
-end
-
 """
     $(TYPEDSIGNATURES)
 
@@ -312,128 +95,37 @@ Keyword arguments:
 All other keyword arguments are forwarded to `HomotopyContinuation.solver_startsystems`.
 """
 function MTK.HomotopyContinuationProblem(
-        sys::NonlinearSystem, u0map, parammap = nothing; eval_expression = false,
-        eval_module = ModelingToolkit, warn_parametric_exponent = true, kwargs...)
+        sys::NonlinearSystem, u0map, parammap = nothing; kwargs...)
+    prob = MTK._safe_HomotopyContinuationProblem(sys, u0map, parammap; kwargs...)
+    prob isa MTK.HomotopyContinuationProblem || throw(prob)
+    return prob
+end
+
+function MTK._safe_HomotopyContinuationProblem(sys, u0map, parammap = nothing; kwargs...)
     if !iscomplete(sys)
         error("A completed `NonlinearSystem` is required. Call `complete` or `structural_simplify` on the system before creating a `HomotopyContinuationProblem`")
     end
-
-    dvs = unknowns(sys)
-    # we need to consider `full_equations` because observed also should be
-    # polynomials (if used in equations) and we don't know if observed is used
-    # in denominator.
-    # This is not the most efficient, and would be improved significantly with
-    # CSE/hashconsing.
-    eqs = full_equations(sys)
-
-    polydata = map(eqs) do eq
-        data = PolynomialData()
-        process_polynomial!(data, eq.lhs, dvs)
-        process_polynomial!(data, eq.rhs, dvs)
-        data
+    transformation = MTK.PolynomialTransformation(sys)
+    if transformation isa MTK.NotPolynomialError
+        return transformation
     end
-
-    has_parametric_exponents = any(d -> d.has_parametric_exponent, polydata)
-
-    all_non_poly_terms = mapreduce(d -> d.non_polynomial_terms, vcat, polydata)
-    unique!(all_non_poly_terms)
-
-    var_to_nonpoly = Dict{BasicSymbolic, PolynomialTransformationData}()
-
-    is_poly = true
-    transformation_err = nothing
-    for t in all_non_poly_terms
-        # if the term involves multiple unknowns, we can't invert it
-        dvs_in_term = map(x -> occursin(x, t), dvs)
-        if count(dvs_in_term) > 1
-            transformation_err = MultivarTerm(t, dvs[dvs_in_term])
-            is_poly = false
-            break
-        end
-        # we already have a substitution solving for `var`
-        var = dvs[findfirst(dvs_in_term)]
-        if haskey(var_to_nonpoly, var) && !isequal(var_to_nonpoly[var].term, t)
-            transformation_err = MultipleTermsOfSameVar([t, var_to_nonpoly[var].term], var)
-            is_poly = false
-            break
-        end
-        # we want to solve `term - new_var` for `var`
-        new_var = gensym(Symbol(var))
-        new_var = unwrap(only(@variables $new_var))
-        invterm = Symbolics.ia_solve(
-            t - new_var, var; complex_roots = false, periodic_roots = false, warns = false)
-        # if we can't invert it, quit
-        if invterm === nothing || isempty(invterm)
-            transformation_err = SymbolicSolveFailure(t, var)
-            is_poly = false
-            break
-        end
-        # `ia_solve` returns lazy terms i.e. `asin(1.0)` instead of `pi/2`
-        # this just evaluates the constant expressions
-        invterm = Symbolics.substitute.(invterm, (Dict(),))
-        # RootsOf implies Symbolics couldn't solve the inner polynomial because
-        # `Nemo` wasn't loaded.
-        if any(x -> MTK.iscall(x) && MTK.operation(x) == Symbolics.RootsOf, invterm)
-            transformation_err = NemoNotLoaded()
-            is_poly = false
-            break
-        end
-        var_to_nonpoly[var] = PolynomialTransformationData(new_var, t, invterm)
-    end
-
-    if !is_poly
-        throw(NotPolynomialError(transformation_err, eqs, polydata))
-    end
-
-    subrules = Dict()
-    combinations = Vector[]
-    new_dvs = []
-    for x in dvs
-        if haskey(var_to_nonpoly, x)
-            _data = var_to_nonpoly[x]
-            subrules[_data.term] = _data.new_var
-            push!(combinations, _data.inv_term)
-            push!(new_dvs, _data.new_var)
-        else
-            push!(combinations, [x])
-            push!(new_dvs, x)
-        end
-    end
-    all_solutions = collect.(collect(Iterators.product(combinations...)))
-
-    denoms = []
-    eqs2 = map(eqs) do eq
-        t = eq.rhs - eq.lhs
-        t = Symbolics.fixpoint_sub(t, subrules; maxiters = length(dvs))
-        # the substituted variable occurs outside the substituted term
-        poly_and_nonpoly = map(dvs) do x
-            haskey(var_to_nonpoly, x) && occursin(x, t)
-        end
-        if any(poly_and_nonpoly)
-            throw(NotPolynomialError(
-                VariablesAsPolyAndNonPoly(dvs[poly_and_nonpoly]), eqs, polydata))
-        end
-
-        num, den = handle_rational_polynomials(t, new_dvs)
-        # make factors different elements, otherwise the nonzero factors artificially
-        # inflate the error of the zero factor.
-        if iscall(den) && operation(den) == *
-            for arg in arguments(den)
-                # ignore constant factors
-                symbolic_type(arg) == NotSymbolic() && continue
-                push!(denoms, abs(arg))
-            end
-        elseif symbolic_type(den) != NotSymbolic()
-            push!(denoms, abs(den))
-        end
-        return 0 ~ num
+    result = MTK.transform_system(sys, transformation)
+    if result isa MTK.NotPolynomialError
+        return result
     end
+    MTK.HomotopyContinuationProblem(sys, transformation, result, u0map, parammap; kwargs...)
+end
 
-    sys2 = MTK.@set sys.eqs = eqs2
-    MTK.@set! sys2.unknowns = new_dvs
-    # remove observed equations to avoid adding them in codegen
-    MTK.@set! sys2.observed = Equation[]
-    MTK.@set! sys2.substitutions = nothing
+function MTK.HomotopyContinuationProblem(
+        sys::MTK.NonlinearSystem, transformation::MTK.PolynomialTransformation,
+        result::MTK.PolynomialTransformationResult, u0map,
+        parammap = nothing; eval_expression = false,
+        eval_module = ModelingToolkit, warn_parametric_exponent = true, kwargs...)
+    sys2 = result.sys
+    denoms = result.denominators
+    polydata = transformation.polydata
+    new_dvs = transformation.new_dvs
+    all_solutions = transformation.all_solutions
 
     _, u0, p = MTK.process_SciMLProblem(
         MTK.EmptySciMLFunction, sys, u0map, parammap; eval_expression, eval_module)
@@ -443,10 +135,11 @@ function MTK.HomotopyContinuationProblem(
     unpack_solution = MTK.build_explicit_observed_function(sys2, all_solutions)
 
     hvars = symbolics_to_hc.(new_dvs)
-    mtkhsys = MTKHomotopySystem(nlfn.f, p, nlfn.jac, hvars, length(eqs))
+    mtkhsys = MTKHomotopySystem(nlfn.f, p, nlfn.jac, hvars, length(new_dvs))
 
     obsfn = MTK.ObservedFunctionCache(sys; eval_expression, eval_module)
 
+    has_parametric_exponents = any(d -> d.has_parametric_exponent, polydata)
     if has_parametric_exponents
         if warn_parametric_exponent
             @warn """