Many neural networks can be derived as optimization dynamics for suitable objective functions. We show that such networks can be designed by repeated transformations of one objective into another with the same fixpoints. We exhibit a collection of algebraic transformations which reduce network cost and increase the set of objective functions that are neurally implementable. The transformations include simplification of products of expressions, functions of one or two expressions, and sparse matrix products (all of which may be interpreted as Legendre transformations); also the minimum and maximum of a set of expressions. These transformations introduce new interneurons which force the network to seek a saddle point rather than a minimum. Other transformations allow control of the network dynamics, by reconciling the Lagrangian formalism with the need for fixpoints. We apply the transformations to simplify a number of structured neural networks, beginning with the standard reduction of...