I'm looking for information on interpolating a surface function p(x,y) based only on estimates of the partial derivatives at points on a grid. Obviously, any such approximation is subject to a constant offset, so assume that p(0, 0) = 0.

Clearly, one could just attempt to build approximations of the partial derivatives and calculate path integrals along the grid, but this is insufficient for my needs because the partial derivatives themselves are approximate and there is no guarantee of path independence of the integral.

I'm looking for leads on more advanced methods that attempt to take into consideration the entire universe of partial derivative information available to identify a probable, smooth surface that aligns to the derivative information as closely as is reasonable.

I could see a minimization approach where one first uses SGD or something similar to find the best constellation of values for p at each grid point by minimizing a cost function based on the given partial derivative information, and then fits to those points using Bezier Surfaces or Hermite polynomials, etc.

But I was wondering if there was some known, specially developed method/algorithm for this task.


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