(1) is substituted either directly (for the energy) or by taking its derivative with respect to atomic positions (to obtain the forces). Let us collect the M unknown coefficients in (1) into a vector x, concatenate all the available data (energies e and forces f ) into the vector y = [e f ], and let L be the linear operator connecting this data vector with the energies of the input configurations, so that y = Le. Note that L consists of two blocks, the upper block is just the identity, and the lower block is the negative differential operator. With this, the regularized least squares problem is linear and can be written as: