A peculiar behavior of `ParametricFunction`

Question

I am trying to use ParametricNDSolve to find a desired value for a parameter in an ODE, and the output seems to misbehave.

First, consider the following example given in the documentation:

sol1 = ParametricNDSolve[{x''[t] - x'[t] + x[t] == 0, x[0] == 1, x'[0] == s}, x, {t, 0, 30}, {s}]
(* ::Output:: *) (*
{x -> ParametricFunction[ <> ]}
*)
root1 = FindRoot[Evaluate[x[s][10] /. sol1], {s, 6}]
(* ::Output:: *) (*
{s -> 1.40296}
*)

Secondly, my equation with the Automatically chosen method:

sol2 = ParametricNDSolve[{X'[t] == t + (Sqrt[3] / Pi) Log[a/(1 - a)] Abs[t X[t]] - X[t], X[0] == 1}, {X}, {t, 0, 2}, a]
(* ::Output:: *) (*
{x -> ParametricFunction[ <> ]}
*)

root2 = FindRoot[Evaluate[X[a][2] /. sol2] == 3, {a, .5}]
(* ::Output:: *) (*
[!] InterpolatingFunction: Input {2} lies outside the range of data... (InterpolatingFunction::dmval)
{a -> 0.812787}
*)

In which I don't understand how the range is exceeded. Furthermore, specifying a method in the call yields:

sol3 = = ParametricNDSolve[{X'[t] == t + (Sqrt[3] / Pi) Log[a/(1 - a)] Abs[t X[t]] - X[t], X[0] == 1}, {X}, {t, 0, 2}, a, Method -> {"TimeIntegration" -> "ExplicitEuler"}]
(* ::Output:: *) (*
{x -> ParametricFunction[ <> ]}
*)

X[.7][2] /. sol3
(* ::Output:: *) (*
2.01701
*)

root3 = FindRoot[Evaluate[X[a][2] /. sol3] == 3, {a, .5}]
(* ::Output:: *) (*
[!] ParametricNDSolve: Encountered invalid NDSolve`SensitivityMethod method data object at point t=0.` (ParametricNDSolve::mdata)
[!] FindRoot: At {a}={0.5}, function value {-3+ParametricFunction[1,Internal`Bag[<1>],1,1,False,{{a$71966},<<5>>,{0}},{NDSolve`base$71973,NDSolve`NDSolveParametricFunction[0,{ParametricNDSolve,Internal`Bag[<2>],None,ParametricNDSolve},{{{<<9>>},{<<9>>}},{0,{<<3>>},{<<2>>},{<<3>>}},None,{{<<7>>},{<<1>>},None,{}}},{X},<<4>>,{Cache->True,CacheTableLength->19,CacheTableWidth->7,CacheKeyMaxBytes->1000000,CacheResultMaxBytes->1000000,KeyComparison->None,ResultComparison->LessEqual},{},<<1>>]}][0.5][2]} is not a {1} dimensional list of numbers  (FindRoot::nlnum)
{a -> 0.5}
*)

X[.7][2] /. sol3
(* ::Output:: *) (*
ParametricFunction[ <> ][0.7][2]
*)

Note that t=2 is indeed in the interpolation range, and the call to the ParametricFunction instance fails after the unsuccessful FindRoot run.

How to understand this behavior, and is there a current workaround if it's a bug?

---

Related:

1. ParametricFunction from ParametricNDSolveValue changes when evaluated?
2. Vector ParametricNDSolve and FindRoot interaction
3. Issue in ParallelTable after evaluating another function using NDSolve and FindRoot

Answer 1

Evaluated -> False

Clear[sol2, root2];
sol2 = ParametricNDSolve[{X'[t] == 
    t + (Sqrt[3]/Pi) Log[a/(1 - a)] Abs[t X[t]] - X[t], 
   X[0] == 1}, {X}, {t, 0, 2}, a]
root2 = FindRoot[(X[a][2] /. sol2) == 3, {a, .5}, Evaluated -> False]

{a -> 0.812787}

Answer 2

Without Method -> {"TimeIntegration" -> "ExplicitEuler"} try more direct solution using ParametricNDSolveValue(only return X[2])

X3 = ParametricNDSolveValue[{X'[t] == t + (Sqrt[3]/Pi) Log[a/(1 - a)] Abs[t X[t]]- X[t], X[0] == 1}, 
X[2] , {t, 0, 2}, a  ]
root3 = FindRoot[ X3[a]   == 3, {a, 1/2 }]
(* {a -> 0.812787} *)

NMinimize evaluates without message " Input value {2} lies outside..."

NMinimize[ (X3[a]  - 3)^2, a]
(*{6.10426*10^-16, {a -> 0.812787}}  *)

Answer 3

and is there a current workaround

Some strange evaluation order? Here is a workaround

sol2 = ParametricNDSolve[{X'[t] == t + (Sqrt[3]/Pi) Log[a/(1 - a)] Abs[t X[t]] - X[t], 
    X[0] == 1}, {X}, {t, 0, 2}, a];
f[a_] := X[a] /. sol2
root2 = FindRoot[Evaluate[f[a][2] /. sol2] == 3, {a, .5}]

No beep and no warnings and no range exceeded messages.

V 13.1 on windows 10

Screen shot. No messages.