Differential Equation Parameter Estimation Using Only A Subset of Variables

Specific Domains Numerics
1

Hi,

I have a DAE problem with 37 variables - I only have data for 12 of those variables -
I would like to run parameter estimation with an L2Loss function based only on the 12 variables for which I actually have data. The examples in the Differential Equations Tutorial are very clear, but they seem to assume that data is available for all variables (Lokta-Volterra, simulates data for two variables, Lorenz for 3) - and in the optimization we use as our input the problem and solver. So my question is:
Is there a direct way to specify the subset of the solution to optimize parameters?
My only thought has been to simulate data from the equations and set weight = 0 for the costfunction on all variables that don’t have data, but maybe there is a better way?

Thanks,

A

3 Likes
2

If you use the save_idxs argument, the ODESolution only has the indices you chose, and thus only calculates the loss with those.

3

Awesome - that’s a great solution. thanks for the idea/tip :slight_smile:

4

Hi Chris -
Thanks for all the help.
My system of DAE is non-linear, so I am assuming the best pacakge to use is NLopt for parameter estimation. However, when I try to follow your tutorial I get an error about dt being set to NaN.

using Distributed
@everywhere using DifferentialEquations
using NLopt
using DiffEqParamEstim
using RecursiveArrayTools

function f(du,u,p,t)
  dx = p[1]*u[1] - u[1]*u[2]
  dy = -3*u[2] + u[1]*u[2]
end

u0 = [1.0;1.0]
tspan = (0.0,10.0)
p = [1.5]
prob = ODEProblem(f,u0,tspan,p)
sol =  DifferentialEquations.solve(prob,Tsit5())

t = collect(range(0,stop=10,length=200))
randomized = VectorOfArray([(sol(t[i]) + .01randn(2)) for i in 1:length(t)])
data = convert(Array,randomized)

obj = build_loss_objective(prob,Tsit5(),L2Loss(t,data),maxiters=10000)


opt = Opt(:GN_ESCH, 1)
min_objective!(opt, obj)
lower_bounds!(opt,[0.0])
upper_bounds!(opt,[5.0])
xtol_rel!(opt,1e-3)
maxeval!(opt, 100000)
(minf,minx,ret) = NLopt.optimize(opt,[1.3])

The repeated error is:

Warning: Automatic dt set the starting dt as NaN, causing instability.
└ @ OrdinaryDiffEq /opt/julia/packages/OrdinaryDiffEq/KqOpn/src/solve.jl:376
┌ Warning: Instability detected. Aborting
└ @ DiffEqBase /opt/julia/packages/DiffEqBase/3gIPB/src/integrator_interface.jl:162

and the result is

(0.037901689333110125, [1.3], :MAXEVAL_REACHED)

Any idea why will this happen?
I also tried to use the JuMP warpper, but apparemty JuMP 0.19 is not compatibel with NLopt.

thanks,

A

5

Bad parameters can cause a divergence, which gives an Inf loss, and the optimizer will keep going. So that repeated error isn’t harmful. Increase the number of evals to keep optimizing more.

6

Thanks, I’ll try that -
A

7

The other question in the same general subject is if I have a DAE system with callbacks
would something in this general terms work?

# Example Callbacks
condition(u,t,integrator) = t in ([21, 42, 63, 84])

function affect!(integrator)
    integrator.u[1] += Dose
    integrator.u[11] = (4*Dose+integrator.u[11]*(integrator.u[1]+integrator.u[2]+(integrator.u[3]+
                integrator.u[4]+integrator.u[5])*integrator.u[16]))/(Dose+(integrator.u[1]+
            integrator.u[2]+(integrator.u[3]+integrator.u[4]+integrator.u[5])*integrator.u[16]))
end



condition1(u,t,integrator) = u[19] < 8
affect1!(integrator) = terminate!(integrator)
  
cb = DiscreteCallback(condition,affect!)
cb1 = DiscreteCallback(condition1, affect1!)
cbset = CallbackSet(cb1,cb)



# Example model
#p=is my parameters a vcetor of 36 parameters
MM = zeros(21,21)
[MM[i,i] = 1 for i in 1:15];
t = 0.0:1.0:100
tspan = (0.0,100)
ODESystem = ODEFunction(PKPD4;mass_matrix=MM)
model_ode(p_) = ODEProblem(ODESystem, IC, tspan,p_) 
solve_model(mp_) = DifferentialEquations.solve(model_ode(mp_), Rodas5(),saveat=1, callback=cbset, tstops=[21,42,63,84])
mock_data = Array(solve_model(p))


### Building loss function (here is where I am most confused)
###Is it here where the callbacks would fit in the loss_objective function?
### I am having trouble figuring this out

loss_objective(mp_, dat) = build_loss_objective(model_ode(mp_), Rodas5(), callback=cbset, tstops=[21,42,63,84], L2Loss(t,dat),maxiters=100000)
juobj(args...) = loss_objective(args, mock_data)(args)



# JuMP model to get parameters
### What algorithm do you recommend for large non-linear problems? I am using :GN_ISRES, but maybe there are better options?

jumodel = Model()
JuMP.register(jumodel, :juobj, 36, juobj, autodiff=true)
@variables jumodel begin
    #p1 to 36
end
@NLobjective(jumodel, Min, juobj(CL_adc, CLD_adc, V1_adc, V2_adc, Kdec, e_adc, P_adc, D_adc,
    Kon_adc, Koff_adc, Agtot, Kint_ag, Kdeg, KDiff_drug_Tumor, Kout_drug,
    Kon_tub, Tub, Koff_tub, e_drug, P_drug, D_drug,
    CL_drug, CLD_drug, V1_drug, V2_drug, 
    Vmax, τ, Kgex, Kglin, ψ, KKillmax, KC50, R_cap,R_kro, c, RCell))
setsolver(jumodel, NLoptSolver(algorithm=:GN_ISRES))

From what you see in this sort of pseudo code - Does it make sense to you? I’ve been trying to test it, but it takes long long time to run, so I am honestly not sure if my code is failing or its just slow.
I did not use the callbacks in my test to make things simpler on first pass.
The code structure seems to make sense (at least to me), but I am concerned I am not inputing the mass matrices or callbacks properly into the loss function.

Thanks,

A

8

Yes it makes sense. However, global optimizers are quite inefficient in how they search the space, so while it will give you good parameters, it’s going to have to solve that DAE a few hundred thousand times

9

I see-

Do you have any recommendation to do any other way? My fear is that if I use a non-global optimizer to narrow the parameter space all I’ll achieve will be to find a local minima. I read in your tutorial that using the two-stage solution method, before the build loss function can work to save time. Will this method still get me close to optimal parameters with non linear solver, or will just stop at local?

Also how do you pick the most appropriate algorithm for NLopt? I have gone through the NLOpt Julia docs as well as the NLopt proper, but I can’t find recommendations on when to choose different algorithm (except for yours recommending :GN_IRES, as the most robust to initial conditions, but this also seems to be the slower. Are there algorithms more appropriate to let’s say stiff problems vs more shallow ones?

Thanks so much!

A

10

That method is specific for ODEs though. A DAE one would be good to add.

Global optimization is just a slow process. You can keep increasing the speed of your DAE, decrease the spot where you declare divergence to drop bad parameters quicker, put a smaller bounding box on the possible parameters, etc to increase the speed. There’s tons that you can do for a specific problem but it does require work to find out what is best.

11

What do you mean the spot where you declare divergence? Would that be the tolerance?
How do you adjust something like that?

I’ve been trying to bound the parameters, but 36 is still lots.

Thanks!

A

12

You might want to look into parallelizing this search somehow.

The unstable_check argument lets you give a function for how the divergence is checked for. Normally it looks for Inf or NaN, and then that warning is given and those parameters are rejected. But if you know your system shouldn’t have concentrations higher than 20,000 for example, you can make a check function that looks for that and shave off some time steps.

13

So something g like this?

function checkconc(u,integrator)
   If integrator.u[1]>20000
       return true
   end
end

And then calls this function

unstable_check=checkconc

Inside the odeProblem definition or when we are actually solving the problem?

Thanks,

A

14

You have to use the documented form: http://docs.juliadiffeq.org/latest/basics/common_solver_opts.html#Miscellaneous-1

1 Like
15

Ok
Thanks for pointing me to the right place :slight_smile:

A

16

Hi Chris,
I have this working - but I have noticed one last issue that I amnot sure how to incorporate to my loose function.
I am using L2 which i think is pretty standard to find the parameters that minimize the differences between my data and my DAE solution.
However, while the scale of u1 is in untis or u16 in e-3, u19 is on the scale of e5 - I assume this makes the parameters to preferentially fit the data of e19 to minimize L2 - neglecting the fit of u1 or u16
Do you know of a way to incorporate a scaling factor that scale by the maximum of each u in L2 or a loss function that might be a better fit for this clearly different scales?

Thanks

ale

17

you can add weights to the L2, or define some more in depth cost function which weights terms by the maximum of u.

18

So something like this?

w=fill(0.0,21)
for i in 1:21
    w[i]=maximum(mock_data[i,:])
end
w=sqrt.(1 ./ w)

loss_objective(mp_, dat) = build_loss_objective(model_ode(mp_), Rodas5(), L2Loss(t,dat;data_weight=w),maxiters=100000)
juobj(args...) = loss_objective(args, mock_data)(args)

Thanks,

A

19

Yes

20

One, I think last question.
My variables are not necessarily measured at the same time -
i may have u1 measured at 3,6 and 9 days and u4 at 5,10,and 15 days for example.
Is there a way to incorporate the heterogeneous times of measurement into the L2loss function?

Thanks,

A