ArrheniusOpt

This repo is used to validate the ability of kinetics parameter optimization using Arrhenius.jl and DiffEqFlux.

1. Adjoint sensitivity

By defining adjoint state as:

$$ \boldsymbol a(t) := -\frac{\partial L}{\partial \boldsymbol u(t)} $$

one can get the evolution ODE for the adjoint state:

$$ \frac{d\boldsymbol a(t)}{dt} = -\boldsymbol a(t) \frac{\partial f(\boldsymbol u(t),\boldsymbol \theta,t)}{\partial \boldsymbol u} $$

Similarly, denoting the adjoint states for parameters and times:

$$ \boldsymbol a_\theta(t) := -\frac{\partial L}{\partial \boldsymbol \theta(t)} \ \boldsymbol a_t(t) := \frac{\partial L}{\partial t(t)} $$

And finally one can get the parameters' gradients against loss function:

$$ \frac{\partial L}{\partial \boldsymbol \theta} = \int_{t_1}^{t_0} \boldsymbol a(t) \frac{\partial f(\boldsymbol u(t),\boldsymbol \theta,t)}{\partial \boldsymbol \theta} dt $$

The adjoint sensitivity method is implemented in folder example/Robertson with three different variants:

• backsolveAdjoint.jl: Direct integration of the augmented adjoint states

  $$ \boldsymbol a_{aug}(t) = [\boldsymbol u(t), \boldsymbol a_u(t), \boldsymbol a_\theta(t), \boldsymbol a_t(t)] $$

• interpolatingAdjoint.jl: Backward solving $\boldsymbol u(t)$ is hard for stiff ODEs, so this method interpolates the forward computed $\boldsymbol u(t)$ in the backward process.

• discreteAdjoint.jl: Interpolations on high-dimensional $t\to\boldsymbol u(t)$ is time consuming. This method use the discrete $\boldsymbol u(t_{i+1})$ as a simplification of $\boldsymbol u(t), t\in[t_{i+1},t_{i}]$ when backward solving the augmented states.