# Why does DiffEqFlux.NeuralODEMM require a constraint equation and a mass matrix?

Hello everyone!

I’m looking at the “Enforcing physical constraints via universal differential-algebraic equations” example (Enforcing Physical Constraints via Universal Differential-Algebraic Equations · DiffEqFlux.jl), which trains a neural ordinary differential equation to solve a stiff ODE with constraints. The model is defined as follows:

``````model_stiff_ndae = NeuralODEMM(nn_dudt2, (u, p, t) -> [u + u + u - 1],
tspan, M, Rodas5(autodiff=false), saveat = 0.1)
``````

As best I can tell, the mass matrix `M = [1. 0 0 // 0 1. 0 // 0 0 0]` and the constraint equation `(u, p, t) -> [u + u + u - 1]` convey the same information: that the state variables of the ODE must always sum to 1. Why are both supplied in this example?

Thank you!

The algebraic equation isn’t well-defined without the MM since it can be scaled.

@ChrisRackauckas Thank you, that makes sense. Can you also elaborate on the difference between the `NeuralDAE` (DiffEqFlux.jl/neural_de.jl at master · SciML/DiffEqFlux.jl · GitHub) and `NeuralODEMM` (DiffEqFlux.jl/neural_de.jl at master · SciML/DiffEqFlux.jl · GitHub) functions? I see that the former only requires specification of a constraint function, whereas the latter also needs the mass matrix. Does having the mass matrix enable more efficient calculation of adjoints or something?

One is semi-explicit while the other is fully implicit. NeuralDAE itself is a bad architecture and the MM form is more stable.