Multidimensional fit and interpolation for scattered unstructured data

Question

I have data structured like {{{i,j},f(i,j)},...}}}. In my actual data the relationship between f(i,j) and j,i is unknown and the goal is to get an estimated f(i,j).
Example data:
list = Flatten[
  Table[{{i + 0.1 Random[], j + 0.1 Random[]}, 10*i*j + Random[]}, {i,
     0, 1, 0.1}, {j, 0, 1, 0.1}], 1]

How do I find a good estimation of f(i,j)?
As a bonus, I would like to find an interpolating function that can predict f(i,j) for intermediate values of i,j not present in the data.

flinty · Accepted Answer

If you have a model for $f$ then you'd go down the fitting route with NonlinearModelFit, or you could fit a plane with ResourceFunction["PlaneOfBestFit"]. However, you could also use Predict here as I will show below:
pf = Predict[Rule @@@ list, Method -> "NeuralNetwork"];
Show[
 Plot3D[pf[{x, y}], {x, 0, 1}, {y, 0, 1}, PlotStyle -> Opacity[.25]],
 ListPointPlot3D[Flatten /@ list]
]

Using Method->"GaussianProcess" also produces a good fit. If you have a lot of data, it is important to avoid over-fitting. We can divide the data into a training set and a validation set in a 70% to 30% ratio by random sample, and we can use the validation data to ensure the predictor isn't over-fitting the data (see cross-validation).
list = Flatten[
   Table[{{i + 0.1 Random[], j + 0.1 Random[]}, 
     10*i*j + Random[]}, {i, 0, 1, 0.05}, {j, 0, 1, 0.05}], 1];

(* divide the data into 70% training and 30% cross-validation *) 
{training, validation} = 
  TakeDrop[#, Round[Length[#]*0.7]] &@RandomSample[Rule @@@ list];

pf = Predict[training, Method -> "GaussianProcess", 
   ValidationSet -> validation];

Show[
 Plot3D[pf[{x, y}], {x, 0, 1}, {y, 0, 1}, PlotStyle -> Opacity[.25]],
 ListPointPlot3D[Flatten /@ list]
 ]
pm = PredictorMeasurements[pf, validation];
pm["RSquared"]
pm["ComparisonPlot"]

By using PredictorMeasurements on the validation set, we can gauge how well the fit is generalizing to unseen data:

Multidimensional fit and interpolation for scattered unstructured data

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