# Pros and cons of multivariate interpolation techniques for scattered data?

I have a numerical simulation $$f$$ that takes 6 input parameters $$\mathbf x = x_1, x_2, \ldots x_6$$. I have randomly selected $$25,000$$ random combinations of these inputs and calculated $$f(\mathbf x)$$. The output of the simulation is about 25 numbers $$\mathbf y = y_1, y_2, \ldots y_{25}$$ (although for the sake of simplicity we could pretend I am only computing $$y_1$$).

I am now trying to interpolate this function 6-dimensional function. I hold out a model, train on the remaining models, and test on the held-out model.

I have tried various schemes:

To my surprise, linear interpolation has outperformed the rest. (To be clear, I have tried various data preprocessing steps to ensure the data are normalized, etc., as well as several hours changing the parameters of each method.)

Is this what I should have expected? Is there another technique that may work better?

I would like to note that I have apparently failed to find a method that can do cubic interpolation with high dimensional input, which I would imagine to be a promising route. (SciPy at least is limited to 2D functions.)