Abstract: Disclosed is a method of generating a functional singularity at a point or collection of points. The method may include determining a relationship between one or more parameters associated with a physical structure and a spatial gradient of field values of at least one of electromagnetic energy, sound energy, particle beam, or water waves manipulated by the physical structure, configuring, according to the relationship, the spatial gradient of field values to represent a functional singularity at a point, performing backpropagation using the spatial gradient of field values to obtain design parameters corresponding to values for the one or more parameters that achieve the functional singularity at the point, and producing a physical structure having the design parameters.