# Param fit for ODE, starting from 0 or restarting from intermediate time points give different results

My problem is the following. I have an ODE with some parameters to estimate, based on some data. In this example however I will generate the data from the ode itself to make some test.
I take an example from LotkaVolterra

``````using DifferentialEquations

function lotka_volterra!(du, u, p, t)
x, y = u
α, β, δ, γ = p
du[1] = α*x - β*x*y
du[2] = -δ*y + γ*x*y
end

u0 = [1.0, 1.0]
t_span = (0.0,10.0)
t_data = 0.0:1:10.0
data_size = length(t_data)
true_p = [1.5, 1.0, 3.0, 1.0]

true_prob = ODEProblem(lotka_volterra!,u0,t_span)
true_sol = solve(true_prob,Tsit5(),p=true_p,saveat=t_data,abstol=1e-12)
data = true_sol.u
``````

so now I have the data to fit.
One option is to write a loss function that takes as input the parameter `p` and gives some kind of error (like L2) of the solution vs the data that I saved. However I would like to test also the following alternative strategy:
divide the time interval onto sub-time spans, corresponding to times between observations. In each of this problem I can setup an ode, that has an initial condition = the data at the beginning on that interval. Then I write a loss function that takes `p` as an input an runs all the sub/problems and compute the error again by testing each solutions with the data at the end of its own sub-time span.

So before even writing the optimization I test that, using the true parameter `true_p` I should get almost 0 error. I solve this problem by setting up an `EnsembleProblem` since it looks like is the best way to do (also parallelism, more on it at the end).

``````t_span_interval = [(t_data[i],t_data[i+1]) for i in 1:data_size-1]
prob_intervall = ODEProblem(lotka_volterra!,u0,t_span_interval[1],true_p)

function prob_func(prob,i,repeat)
ODEProblem(prob.f,data[i],t_span_interval[i],prob.p)
end
ensemble_prob = EnsembleProblem(prob_intervall,prob_func=prob_func)
ensemble_sol = solve(ensemble_prob,Tsit5(),abstol=1e-12,trajectories=data_size-1,
save_everystep=false,save_start=true,save_end=true)

data_ensemble = [ensemble_sol.u[i][end] for i in 1:data_size-1]
data_ensemble .- data[2:data_size]

``````

Running it, this is the output:

``````julia> data_ensemble .- data[2:data_size]
10-element Vector{Vector{Float64}}:
[2.7065844739659894e-5, -1.3964528888632532e-5]
[0.003105915036464957, -0.0024319585862495607]
[8.646330812744907e-5, 0.00036598671302412455]
[-6.986510828399517e-5, 4.878692306975463e-6]
[-0.0004903808993592662, 0.000827828489636806]
[-0.006222412625121798, 0.0055218332192099595]
[0.0011076821208162446, 0.00037821974241425416]
[-5.89828264159209e-5, -5.9286058622209925e-5]
[0.002522886083798692, 0.003895955430558651]
[-0.00044247516121154185, -0.0003218457894433868]
``````

which looks like is way too big. I set the tol of the solvers to 1e-12 just to see if was a numerical error but it doesn’t look like it. So where is the issue?

Why do this? Just to give it a try. I am no expert in data fitting or optimization, so perhaps this is very wrong. However I would like to see what happens and also exploit the fact that this form of method is highly parallelizable since odes in different intervals are independent.

edit just in case there is something terribly wrong on doing this
A possible error function would be

``````function loss(p)
tmp_prob = remake(prob_intervall,p = p)
tmp_ensemble_prob = EnsembleProblem(tmp_prob,prob_func=prob_func)
tmp_sol = solve(tmp_ensemble_prob,trajectories=data_size-1,
save_everystep=false,save_start=false,save_end=true)
return sum([sum(abs2,tmp_sol.u[i][end].-data[i+1]) for i in 1:data_size-1 ])
end
``````
1 Like

just a minor update. Apart from the original problem, that I didn’t solve, I also tried finally to optimize the parameters by using the loss function described above.
The process is extremely slow compared to perform optimization using a standard loss function where you solve again in the entire interval (tested on a very large problem, size 10k more or less). Also apparently I cannot take advantage of the parallelism of the ensemble. If I look at htop during the optimization process I see all the corse been activated for a brief period of time, then again the usage goes to zero, then again up ecc.ecc…

I don’t quite get what your question is. Can you restate it?

sure thing! sorry for my messed post.
Anyway, let me describe my problem:
I have an ODE in the time interval `t_span`. I can solve it with `DifferentialEquations.jl` and it’s all good.
Now I tried to do the following: save the values of the solutions at certain points in time, described by `t_data` and saved in the variable `data`. Now consider many subproblems, each following the same equation as the original one, but are considered only in times between two consecutive points of `t_data`. each subproblem is also considered to have an initial condition equal to element of `data` corresponding to the same time point. So basically what we are doing is splitting the time `t_span` into subpieces and solving again the ode, by using as the initial condition (for each sub problem) the actual value that the solution assumed in that time point.

Now, if we measure the error between the real (real meaning solved in the whole interval ad once) and each subsolution (in sub intervals) they differ quite a lot, even by setting a tolerance quite small as `1e-12`.
I post the full code here all in one block just to me more clear:

``````using DifferentialEquations

function lotka_volterra!(du, u, p, t)
x, y = u
α, β, δ, γ = p
du[1] = α*x - β*x*y
du[2] = -δ*y + γ*x*y
end

u0 = [1.0, 1.0]
t_span = (0.0,10.0)
t_data = 0.0:1:10.0
data_size = length(t_data)
true_p = [1.5, 1.0, 3.0, 1.0]

true_prob = ODEProblem(lotka_volterra!,u0,t_span)
true_sol = solve(true_prob,Tsit5(),p=true_p,saveat=t_data,abstol=1e-12)
data = true_sol.u

t_span_interval = [(t_data[i],t_data[i+1]) for i in 1:data_size-1]
prob_intervall = ODEProblem(lotka_volterra!,u0,t_span_interval[1],true_p)

function prob_func(prob,i,repeat)
ODEProblem(prob.f,data[i],t_span_interval[i],prob.p)
end
ensemble_prob = EnsembleProblem(prob_intervall,prob_func=prob_func)
ensemble_sol = solve(ensemble_prob,Tsit5(),abstol=1e-12,trajectories=data_size-1,
save_everystep=false,save_start=true,save_end=true)

data_ensemble = [ensemble_sol.u[i][end] for i in 1:data_size-1]
data_ensemble .- data[2:data_size]
``````

which outputs

``````julia> data_ensemble .- data[2:data_size]
10-element Vector{Vector{Float64}}:
[2.7065844739659894e-5, -1.3964528888632532e-5]
[0.003105915036464957, -0.0024319585862495607]
[8.646330812744907e-5, 0.00036598671302412455]
[-6.986510828399517e-5, 4.878692306975463e-6]
[-0.0004903808993592662, 0.000827828489636806]
[-0.006222412625121798, 0.0055218332192099595]
[0.0011076821208162446, 0.00037821974241425416]
[-5.89828264159209e-5, -5.9286058622209925e-5]
[0.002522886083798692, 0.003895955430558651]
[-0.00044247516121154185, -0.0003218457894433868]
``````

Because when using a PI-controller for stepsize adaptivity you’re not getting the same time steps. You probably also want to decrease the `reltol=1e-12`.

3 Likes