We recently released a new manuscript Semi-Explicit Neural DAEs: Learning Long-Horizon Dynamical Systems with Algebraic Constraints where we showed a way to develop neural networks where any arbitrary constraint function can be directly imposed throughout the evolution equation to near floating point accuracy. However, in true academic form it focuses directly on getting to the point about the architecture, but here I want to elaborate about the mathematical structures that surround the object, particularly the differential-algebraic equation (DAE), how its various formulations lead to the various architectures (such as stabilized neural ODEs), and elaborate on the other related architectures which haven’t had a paper yet but how you’d do it (and in what circumstances they would make sense).