# Using JuMP with DiffEqParamEstim

JuMP is a domain-specific modeling language for mathematical optimization embedded in Julia.

using OrdinaryDiffEq, DiffEqParamEstim, JuMP, NLopt, Plots

Let's define the Lorenz equation to use as our example

function g(du,u,p,t)
σ,ρ,β = p
x,y,z = u
du[1] = dx = σ*(y-x)
du[2] = dy = x*(ρ-z) - y
du[3] = dz = x*y - β*z
end

Let's get a solution of the system with parameter values σ=10.0 ρ=28.0 β=8/3 to use as our data. We define some convenience functions model_ode (to create an ODEProblem) and solve_model(to obtain solution of the ODEProblem) to use in a custom objective function later.

u0 = [1.0;0.0;0.0]
t = 0.0:0.01:1.0
tspan = (0.0,1.0)
model_ode(p_) = ODEProblem(g, u0, tspan,p_)
solve_model(mp_) = OrdinaryDiffEq.solve(model_ode(mp_), Tsit5(),saveat=0.01)
mock_data = Array(solve_model([10.0,28.0,8/3]))

Now we define a custom objective function to pass for optimization to JuMP using the build_loss_objective described above provided by DiffEqParamEstim that defines an objective function for the parameter estimation problem.

loss_objective(mp_, dat) = build_loss_objective(model_ode(mp_), Tsit5(), L2Loss(t,dat))

We create a JuMP model, variables, set the objective function and the choice of optimization algorithm to be used in the JuMP syntax. You can read more about this in JuMP's documentation.

juobj(args...) = loss_objective(args, mock_data)(args)
jumodel = Model()
JuMP.register(jumodel, :juobj, 3, juobj, autodiff=true)
@variables jumodel begin
σ,(start=8)
ρ,(start=25.0)
β,(start=10/3)
end
@NLobjective(jumodel, Min, juobj(σ, ρ, β))
setsolver(jumodel, NLoptSolver(algorithm=:LD_MMA))

Let's call the optimizer to obtain the fitted parameter values.

sol = JuMP.solve(jumodel)
best_mp = getvalue.(getindex.((jumodel,), Symbol.(jumodel.colNames)))

Let's compare the solution at the obtained parameter values and our data.

sol = OrdinaryDiffEq.solve(best_mp |> model_ode, Tsit5())
plot(getindex.(sol.(t),1))
scatter!(mock_data, markersize=2)