While for 1D interpolation this could be trivial (if the value is outside the grid then fail) it becomes exponentially tricky as D increases.

Thus I'd suggest providing some extrapolation routines attached to the dimensionality of the grid.

Even so, there are two different situations:

• Complete extrapolation
• d-n extrapolation (so, only d-n variables are outside of the region)

I think it would be a good idea to provide some helpers in the second case so that part of the interpolation is provided "ok, the result goes out but the edge is x"

Benchmark interpolation algorithms:

• in Python
• on known functions
• plotting them in 2d
• plotting relative difference heatmap
• measuring performances

Usually increasing complexity is one to one to higher frequencies.

It is said to be required (sometimes)

https://arxiv.org/pdf/1710.03016.pdf (sec. 5.1)