Hessian of function involving integration of ODE

Specific Domains Machine Learning
1

Hi all,
I apologize if it’s a possibly stupid question, but I am trying to calculate the Hessian of a function that internally performs the integration of a differential equation. What I am currently doing is this:

using Zygote, DifferentialEquations
function der!(du, u, p, t)
    du[1] = -p[1] * u[1]
end

function cost_for_hessian(params)
    prob_uode_pred = ODEProblem{true}(der!, [10.0], (0, 10))
    solutions = solve(prob_uode_pred, Tsit5(), p=params, saveat=[10.0],  sensealg=ForwardDiffSensitivity())
    return sum(abs2.((solutions)))
end


sensitivity_analysis_hessian = hessian(p -> cost_for_hessian(p), [2.0])

When I try to execute the code (julia version 1.9.1, Zygote v0.6.61, DifferentialEquations v7.7.0), I get the following error: ERROR: Cannot determine ordering of Dual tags.

Could anyone tell me if what I am trying to do is feasible and how to do it?
Thank you in advance!

2

Use Forward over Reverse, not Zygote.hessian. These examples

make use of that. It’s done in Optimization.jl like this:

Or you can directly use Direct Adjoint Sensitivities of Differential Equations · SciMLSensitivity.jl

3

Thank you!