Have you looked at ApproXD.jl? 
I wrote a gist comparing the functionality of Grid.jl, Dierckx.jl, 
ApproXD.jl and NumericalMath.jl a while ago (here 
<https://github.com/nilshg/LearningModels/blob/master/Test_Interpolations.jl>), 
the ApproXD example in there generalizes easily to three dimensions.

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