Fix dual tag ordering in second order

Fixes #1184. I'm not sure why the other test didn't hit this, but this just generally makes it more robust so now we have a stricter test.

Author: ChrisRackauckas

src/second_order.jl

@@ -8,10 +8,15 @@ function _second_order_sensitivities(loss, prob, alg, sensealg::ForwardDiffOverA
     end
 end
 
+struct SciMLSensitivityTag end
+
+
+
 function _second_order_sensitivity_product(loss, v, prob, alg,
         sensealg::ForwardDiffOverAdjoint,
         args...; kwargs...)
-    θ = ForwardDiff.Dual.(prob.p, v)
+    T = typeof(ForwardDiff.Tag(SciMLSensitivityTag(),eltype(v)))
+    θ = ForwardDiff.Dual{T,eltype(v),1}.(prob.p, ForwardDiff.Partials.(Tuple.(v)))
     _loss = p -> loss(solve(prob, alg, args...; p = p, sensealg = sensealg.adjalg,
         kwargs...))
     getindex.(ForwardDiff.partials.(Zygote.gradient(_loss, θ)[1]), 1)

test/second_order.jl

@@ -34,3 +34,17 @@ H2v = H * v
 
 @test H ≈ H2
 @test Hv ≈ H2v
+
+function lotka!(du,u,p,t)
+    du[1] = dx = p[1]*u[1] - p[2]*u[1]*u[2]
+    du[2] = dy = -p[3]*u[2] + p[4]*u[1]*u[2]
+  end
+
+p = [1.5,1.0,3.0,1.0]; u0 = [1.0;1.0]
+prob = ODEProblem(lotka!,u0,(0.0,10.0),p)
+loss(sol) = sum(sol)
+v = ones(4)
+
+Hv = second_order_sensitivity_product(loss,v,prob,Vern9(),saveat=0.1,abstol=1e-12,reltol=1e-12)
+forward_Hv = ForwardDiff.hessian(p -> sum(solve(prob, Vern9(), p=p,saveat=0.1,abstol=1e-12,reltol=1e-12)), p)*v
+@test Hv ≈ forward_Hv