In this section, we summarize the essentials of PM-DDNN, and we introduce the new functional form for the parametrically managed activation function. Since the discussion of previously published work is just a recap, we refer the readers to ref. 28 A key aspect of the PM-DDNN method is that it treats the DPEM as a sum of two parts, namely, a many-body matrix that is learned by the neural network and a pre-fit diagonal lower-dimensional matrix, 𝚵 ! "# , which, like the DPEM, is stored as a one-dimensional array of length 𝜈, where 𝜈 is the number of unique elements of the symmetric DPEM as given by