interpolate 1D signal depending on 3D parameter space
What you are trying to do is "guess" the behaviour of $d$ given $p_1,p_2,p_3$ without going through a, potentially costly, full evaluation of the expression that generates it.

Is this interpolation? No.

Inteprolation will be handy, to a certain extent, but the problem sounds more like one of classification. If you treat it as classification then the Discrete Fourier Transform will also be useful.

So, given the values of the set of $p_i$'s, is $d$ spiky, bumpy, smooth, other? "Spiky" now becomes a region of $p_i$. It doesn't have to be continuous, it doesn't have to be smooth. All you need to know is that as $p_i$ tends to certain places in this $\mathbb{R}^3$ space, $d$ is more likely to be "Spiky".

How do you decide if something is Spiky, Smooth or Bumpy? You look at its spectrum using the Discrete Fourier Transform. You use the spectrum as a feature that tells you what $d$ looks like. Specifically, sharp waveforms in the time domain give you wide spectra and vice versa. So, a spiky $d$, has a wide spectrum. A smooth slowly varying $d$ might show a few bumps at the low end of the spectrum but then die down quickly at the top end and of course flat $d$s will only have a DC component.

You will still not be able to do a full blown evaluation of $d$ across all combinations of $p_i$. First of all, some combinations of $p_i$ might be nonsensical, which is a good thing because it reduces the size of your search space. This is where interpolation comes in, by assuming that nearby values of $p_i$ lead to qualitatively similar $d$s. This is a judgement call but of course if the expression that leads to $d$ has $p_i$ involved in subtractions in some denominator, then small changes in $p_i$s can produce wild swings in the values of $d$. So, in the end, you might want to produce a "landscape" of $d$ for a quantised grid of $p_i$s.

Here is such a landscape but from another field:



The figure is reproduced from this paper

The authors take Jansen's model which models the behaviour of a population of neurons and want to see what happens when more than one populations are coupled together.

The model has a number of parameters but there is no function that tells you what parameters to plug to the model so that its output oscillates at a particular frequency. The model is capable of a wild repertoire of outputs from smooth, to fast oscillating to spiking to even just intermittent, a few spikes and then a saturated value.

So, the landscape that you see here was necessary to examine the behaviour of the coupled neuronal population when either of them was supposed to be oscillating in different ways.

(And yes, evaluation of Jansen's model for realistic waveforms (e.g. 20-30 seconds) is costly, the model is defined as a set of coupled differential equations which you have to integrate (carefully) across all the different populations you are trying to simulate. This can be very slow, especially when you try to do it at a very high level language with lots of layers between you and the hardware).

Your application is very similar to this but you are not as lucky as only being interested in oscillatory activity which has a well defined spectrum.

So, the challenge here will be how do you quantify the behaviour of $d$ using the right features to characterise the "bumpiness" (or smoothness) of $d$.

Just out of curiosity, what does the expression that lead to $d$ looks like?

Hope this helps.