Iterative completion of PDE solutions

I have been reading this article,
Where they iteratively complete a solution symbolically.
It is possible to do something similar numerical in Julia ?
I want to put in a numerical function and look for a solution that is close to it. Is this possible ? Does such a method already exist somewhere in SciML ?

I’m not sure precisely what you mean, but there are many, many “iterative” solution methods, for many kinds of equations, where you start with an initial guess and converge towards a solution. What kind of equation do you want to solve? Why not just use a standard numerical-PDE method like finite elements (e.g. Gridap.jl), finite differences, MethodOfLines.jl, etcetera?

Sure, you can pose it as an optimization problem. Express the solution say as an orthogonal polynomial, a chebyshev series for example. Then you could express the problem as minimizing the norm of the error between the input function and the solution, subject to the constraints that it collocates zero error in the differential equation at the Chebyshev points.

This would be as an alternative to boundary conditions for example.