Prevent Plus from threading InterpolatingFunction

Question

For an InterpolatingFunction $y:\ \mathbb{R}\to\mathbb{R}^2$, Plus is unaware of this, so when I do any modifications to it in an unevaluated form, it breaks the result

y = Interpolation[Table[{i, RandomReal[{0, 1}, 2]}, {i, 1, 10}]];
y[t]
y[t] + {1, 2}

(* InterpolatingFunction[{{1.,10.}},<>][t]`
{1+InterpolatingFunction[{{1.,10.}},<>][t],2+InterpolatingFunction[{{1.,10.}},<>][t]} *)

I ran into it doing things like:

y = Interpolation[Table[{i, RandomReal[{0, 1}, 2]}, {i, 1, 10}]];
f[t_] := Piecewise[{{ {1, 2} + y[t], t < 5}}];
df[t_] := Evaluate[D[f[t], t]]
df[t]
df[3]

And I can't figure out how to prevent expressions like {1,2} + y[t] from being evaluated while still being able to differentiate them. If Plus just realized that y[t] is $\mathbb{R}^2$ valued, I think that would simplify everything for me. Is there some way to set this property?

Answer 1

Using the InterpolatingFunctionAnatomy package one can implement the threading explicitly.

y = Interpolation[Table[{i, RandomReal[{0, 1}, 2]}, {i, 1, 10}]];

<< DifferentialEquations`InterpolatingFunctionAnatomy`

yy = Interpolation[
  MapThread[
   List, {InterpolatingFunctionGrid[y], {1, 2} + # & /@ 
     InterpolatingFunctionValuesOnGrid[y]}], 
  InterpolationOrder -> 
   First[InterpolatingFunctionInterpolationOrder[y]]]