Boundary element methods can be used to solve differential equations, where for some of the elements (boundary elements), the solution value is given/ fixed. Boundary element methods have been used in ML, example being solving image blending task using poisson equation with given boundary elements.

https://en.wikipedia.org/wiki/Discrete_Poisson_equation

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My question is: What is the effect of sparsity of boundary elements on the solution of the differential equation ?