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I have a 4-parameter family of functions, which I'll refer to as y[a, b, c, d][x]. In this notation, a, b, c, d are the 4 parameters, and y[a, b, c, d] is a function $\mathbb{R}\to\mathbb{R}$.

I would like to implement a Manipulate-based widget, with one slider for each of the 4 parameters, that will show a 2-D plot of y[a, b, c, d][x] vs x for any given setting of the parameters.

The hitch is that the functions y[a, b, c, d] are very time-consuming to compute. Each plot would take about 1 minute to generate, which rules out any interactive exploration.

Luckily, the functions y[a, b, c, d] appear to be rather smooth, so I'm wondering if there is some way to approximate the ensemble with some easier-to-compute "spline-like" functions yApprox[a, b, c, d].

My naive thinking about this begins with a pre-computed mesh of points of the form

{a, b, c, d, x, y[a, b, c, d][x]}

...but I'm not sure how to use such a mesh to generate an ensemble of continuous functions yApprox[a, b, c, d] varying continuously over the parameters a, b, c, d.

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