Let's break this puzzle down into pieces. We need some additional assumptions, such as the age of the plant - a fully mature plant consists of only the tubers since the haulm and root will have dried up to nothing since their job is done. A very young plant will weigh about half a potato which will be the chitted piece with sprouting eyes. So probably what you are looking for is a plant at full green maturity where the tubers are almost finished sizing up and the green part has not started to die down - a maxo plant. I just coined that word to describe the biggest size of the package.
So this maxo plant consists of about 7-10 tubers of various sizes, the green top and the root; as a rule of thumb you can say that the root volume is about the same as the green top otherwise the green top could not be supported in terms of water and nutrient. So if we can find the weight of the top then we also have approx. the weight of the root.
Potato haulms consist of thick and watery stems, one from each eye that decides to expand. A reasonable number of expanded eyes might be 10-12, and these will branch a little as they extend, but not much, certainly not as much as a tomato given its head. So imagine that potato stem sitting in front of you and assess the weight. Most of it is water - scrunch it down into a cube and how many cc do you have, let's say 5x5x5 = 125 cc.
So in kg.:
- 10 tubers (medium Russet) ....... 1.0
- Green top x 12 @ 125cc water .... 1.5
- Root same as top ................ 1.5
Total 4 kilograms.
Edit: to describe the weight over time as the plant grows we need a function which evaluates to half a potato at t=0, 4 kg at t=maxo, and 1 kg at t=maturity. A sine wave won't serve because it grows too fast at the beginning. A sigmoid/exponential would be better for the first part to maxo, but then it needs damping down as the top dries out and the water is recaptured and stored for re-use with the following crop.
There are some articles on line discussing modelling of plant growth, see for example "Dynamical Models of Plant Growth" by Bessonov and Volpert which will give you keys into other work.