# Enforcing Physical Constraints via Universal Differential-Algebraic Equations

As shown in the stiff ODE tutorial, differential-algebraic equations (DAEs) can be used to impose physical constraints. One way to define a DAE is through an ODE with a singular mass matrix. For example, if we make Mu' = f(u) where the last row of M is all zeros, then we have a constraint defined by the right hand side. Using NeuralODEMM, we can use this to define a neural ODE where the sum of all 3 terms must add to one. An example of this is as follows:

using DiffEqFlux, DifferentialEquations, Plots

function f!(du, u, p, t)
y₁, y₂, y₃ = u
k₁, k₂, k₃ = p
du[1] = -k₁*y₁ + k₃*y₂*y₃
du[2] =  k₁*y₁ - k₃*y₂*y₃ - k₂*y₂^2
du[3] =  y₁ + y₂ + y₃ - 1
return nothing
end

u₀ = [1.0, 0, 0]
M = [1. 0  0
0  1. 0
0  0  0]

tspan = (0.0,1.0)
p = [0.04, 3e7, 1e4]

stiff_func = ODEFunction(f!, mass_matrix = M)
prob_stiff = ODEProblem(stiff_func, u₀, tspan, p)
sol_stiff = solve(prob_stiff, Rodas5(), saveat = 0.1)

nn_dudt2 = FastChain(FastDense(3, 64, tanh),
FastDense(64, 2))

model_stiff_ndae = NeuralODEMM(nn_dudt2, (u, p, t) -> [u[1] + u[2] + u[3] - 1],
tspan, M, Rodas5(autodiff=false), saveat = 0.1)
model_stiff_ndae(u₀)

function predict_stiff_ndae(p)
return model_stiff_ndae(u₀, p)
end

function loss_stiff_ndae(p)
pred = predict_stiff_ndae(p)
loss = sum(abs2, Array(sol_stiff) .- pred)
return loss, pred
end

callback = function (p, l, pred) #callback function to observe training
display(l)
return false
end

l1 = first(loss_stiff_ndae(model_stiff_ndae.p))
result_stiff = DiffEqFlux.sciml_train(loss_stiff_ndae, model_stiff_ndae.p,
BFGS(initial_stepnorm = 0.001),
cb = callback, maxiters = 100)

## Step-by-Step Description

using DiffEqFlux, DifferentialEquations, Plots

### Differential Equation

First, we define our differential equations as a highly stiff problem which makes the fitting difficult.

function f!(du, u, p, t)
y₁, y₂, y₃ = u
k₁, k₂, k₃ = p
du[1] = -k₁*y₁ + k₃*y₂*y₃
du[2] =  k₁*y₁ - k₃*y₂*y₃ - k₂*y₂^2
du[3] =  y₁ + y₂ + y₃ - 1
return nothing
end

### Parameters

u₀ = [1.0, 0, 0]

M = [1. 0  0
0  1. 0
0  0  0]

tspan = (0.0,1.0)

p = [0.04, 3e7, 1e4]
• u₀ = Initial Conditions
• M = Semi-explicit Mass Matrix (last row is the constraint equation and are therefore

all zeros)

• tspan = Time span over which to evaluate
• p = parameters k1, k2 and k3 of the differential equation above

### ODE Function, Problem and Solution

We define and solve our ODE problem to generate the "labeled" data which will be used to train our Neural Network.

stiff_func = ODEFunction(f!, mass_matrix = M)
prob_stiff = ODEProblem(stiff_func, u₀, tspan, p)
sol_stiff = solve(prob_stiff, Rodas5(), saveat = 0.1)

Because this is a DAE we need to make sure to use a compatible solver. Rodas5 works well for this example.

### Neural Network Layers

Next, we create our layers using FastChain. We use this instead of Chain because it reduces the overhead making it faster for smaller NNs of <200 layers (similarly for FastDense). The input to our network will be the initial conditions fed in as u₀.

nn_dudt2 = FastChain(FastDense(3, 64, tanh),
FastDense(64, 2))

model_stiff_ndae = NeuralODEMM(nn_dudt2, (u, p, t) -> [u[1] + u[2] + u[3] - 1],
tspan, M, Rodas5(autodiff = false), saveat = 0.1)
model_stiff_ndae(u₀)

Because this is a stiff problem, we have manually imposed that sum constraint via (u,p,t) -> [u[1] + u[2] + u[3] - 1], making the fitting easier.

### Prediction Function

For simplicity, we define a wrapper function that only takes in the model's parameters to make predictions.

function predict_stiff_ndae(p)
return model_stiff_ndae(u₀, p)
end

### Train Parameters

Training our network requires a loss function, an optimizer and a callback function to display the progress.

#### Loss

We first make our predictions based on the current parameters, then calculate the loss from these predictions. In this case, we use least squares as our loss.

function loss_stiff_ndae(p)
pred = predict_stiff_ndae(p)
loss = sum(abs2, sol_stiff .- pred)
return loss, pred
end

l1 = first(loss_stiff_ndae(model_stiff_ndae.p))

Notice that we are feeding the parameters of model_stiff_ndae to the loss_stiff_ndae function. model_stiff_node.p are the weights of our NN and is of size 386 (4 * 64 + 65 * 2) including the biases.

#### Optimizer

The optimizer BFGS is directly passed in the training step (see below).

#### Callback

The callback function displays the loss during training.

callback = function (p, l, pred) #callback function to observe training
display(l)
return false
end

### Train

Finally, training with sciml_train by passing: loss function, model parameters, optimizer, callback and maximum iteration.

result_stiff = DiffEqFlux.sciml_train(loss_stiff_ndae, model_stiff_ndae.p,
BFGS(initial_stepnorm = 0.001),
cb = callback, maxiters = 100)