Discrepancy in DiffEqParamEstim

Specific Domains Optimization (Mathematical)
1

Before I open a bug report on the official repo, just want to make sure I am on the right path. Consider a simple 2 parameter estimation of an ODE.

using DifferentialEquations
using DiffEqParamEstim
using Optim

# some data for reproducibility 
data = [3140, 3232, 3488, 3415, 3192, 3265, 2961, 3548, 4269, 4681, 4415, 4883, 4150, 4394, 5856, 7126, 9130, 9367, 10631, 10737, 11688, 14993, 20665, 23957, 23621, 21338, 22034, 33781, 32013, 39840, 45961, 45359, 46126, 38077, 37410, 39431]

# growth model 
function f(C,params,t)
    r, p = params[1], params[2]
    return r*C^p
end
c0 = 3140n.                 # initial condition
tspan = (0,length(data)-1)  # time span
init_params = [0.5, 0.8]    # initial parameters
prob = ODEProblem(f, c0, tspan, init_params) #sets up the problem 
sol = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8, saveat=1) # get the solution to plot, etc

# lets try to optimize the parameters r and p 
cost_function = build_loss_objective(prob, Tsit5(), L2Loss(tspan, data), maxiters=10000,verbose=false)

cost_function([0.5, 0.2]) # evaluate the cost of these two specific parameter values

The return of build_loss_objective is a function (i.e., cost_function) can be evaluated for different values of parameters r and p, i.e., cost_function([r, p]) which gives you a numeric value. This makes sense.

Now lets use LeastSquaresOptim.jl to solve the optimization using least squares. We will build our cost_function using the provided build_lsoptim_objective(prob, t, data, alg) which is slightly different than build_loss_objective. We have:

cost_function = build_lsoptim_objective(prob, 1, data, Tsit5())
lsp = LeastSquaresProblem(x = [0.5, 0.5], f! = cost_function, output_length = 36)
res = optimize!(lsp, LeastSquaresOptim.Dogleg())

and here is where I found the inconsistency. I had to read the source code in order to figure out the inconsistency. Let me explain:

  • The t argument in the build_lsoptim_objective function sounds like it accepts tspan, i.e., the time span for the ODE problem. This is not the case. Line 10 of the source code suggests the t argument is used only for the saveat parameter in the solver. This is why my cost function has that integer 1.

  • build_lsoptim_objective returns a function, just like build_loss_objective above. Except, it has the signature

function (out, p)
... 
end

where out is a pre-allocated vector to store the results of the ODE in for a given parameter p. This is where the inconsistency is. However, like in the above example, you would expect that cost_function would accept an array of parameters and simply evaluate the cost. That is, cost_function([r, p] should produce a value for for any two parameters r and p… but instead the correct call is cost_function(zeros(length_data), [r, p]).

  • The annoying part here then becomes the argument output_length in the LeastSquaresProblem constructor. For a few hours, I thought the output_length referred to the number of parameters that one is trying to optimize, but infact this is the number of data points in the solution of the ODE.

I hope all that made sense… Kind of seems long. Maybe there is a better way to explain.

2

DiffEqParamEstim was overhauled to be built around Optimization.jl. See the new documentation.

https://docs.sciml.ai/DiffEqParamEstim/stable/getting_started/