Handle conditionals and abs

It's time we finally handle #531, https://github.com/SciML/ModelingToolkit.jl/issues/532, and https://discourse.julialang.org/t/differentiation-method-for-element-wise-abs-function-applied-on-operation-types-in-modelingtoolkit-jl/45867/4 . This PR creates a `ModelingToolkit.ifelse` with derivative fixes for `abs` to make it so standard conditional code can work. While tracing cannot correctly make these operations, this would at least allow directly written symbolic code to handle conditions, so we're at least as good (or bad) as TensorFlow.

Author: ChrisRackauckas

src/ModelingToolkit.jl

@@ -123,6 +123,7 @@ include("systems/reduction.jl")
 
 include("latexify_recipes.jl")
 include("build_function.jl")
+include("extra_functions.jl")
 
 export ODESystem, ODEFunction, ODEFunctionExpr, ODEProblemExpr
 export SDESystem, SDEFunction, SDEFunctionExpr, SDESystemExpr

src/differentials.jl

@@ -180,7 +180,7 @@ derivative(O::Constant, ::Any) = Constant(0)
 # Pre-defined derivatives
 import DiffRules
 for (modu, fun, arity) ∈ DiffRules.diffrules()
-    fun in [:*, :+] && continue # special
+    fun in [:*, :+, :abs] && continue # special
     for i ∈ 1:arity
 
         expr = if arity == 1

src/extra_functions.jl

@@ -0,0 +1,16 @@
+function ifelse end
+@register Base.conj(x)
+@register Base.getindex(x,i)
+@register Base.binomial(n,k)
+@register Base.copysign(x,y)
+
+@register Base.signbit(x)
+ModelingToolkit.derivative(::typeof(signbit), args::NTuple{1,Any}, ::Val{1}) = 0
+
+@register Base.abs(x)
+ModelingToolkit.derivative(::typeof(abs), args::NTuple{1,Any}, ::Val{1}) = ModelingToolkit.ifelse(signbit(args[1]),-one(args[1]),one(args[1]))
+
+@register ModelingToolkit.ifelse(x,y,z)
+ModelingToolkit.derivative(::typeof(ModelingToolkit.ifelse), args::NTuple{3,Any}, ::Val{1}) = 0
+ModelingToolkit.derivative(::typeof(ModelingToolkit.ifelse), args::NTuple{3,Any}, ::Val{2}) = ModelingToolkit.ifelse(args[1],1,0)
+ModelingToolkit.derivative(::typeof(ModelingToolkit.ifelse), args::NTuple{3,Any}, ::Val{3}) = ModelingToolkit.ifelse(args[1],0,1)

src/function_registration.jl

@@ -99,18 +99,10 @@ for fun ∈ [:<, :>, :(==), :&, :|, :div]
     @eval @register $sig
 end
 
-# ifelse
-#@register Base.ifelse(cond,t,f)
-
 # special cases
 Base.:^(x::Expression,y::T) where T <: Integer = Operation(Base.:^, Expression[x, y])
 Base.:^(x::Expression,y::T) where T <: Rational = Operation(Base.:^, Expression[x, y])
 
-@register Base.conj(x)
-@register Base.getindex(x,i)
-@register Base.binomial(n,k)
-@register Base.copysign(x,y)
-
 Base.getindex(x::Operation,i::Int64) = Operation(getindex,[x,i])
 Base.one(::Operation) = 1