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@charlesm93 you keep getting back to adjoint ODEs, which is great. You should have a decent look at the AMICI project here: https://github.com/AMICI-dev/AMICI
they published a performance evaluation of adjoint vs forward: https://journals.plos.org/ploscompbiol/article/file?id=10.1371/journal.pcbi.1005331&type=printable
see figure 2 a. In their example the adjoint method is faster with just 3 parameters in comparison to forward (with 47 states).
So if either the number of parameters or the number of states is large, then the adjoint method is winning. For smallish PK/PD problems with 5 states and 5 parameters forward may actually be fine, but having adjoint ODE in Stan is still a nice thing for the future.
@wds15 Thank you for sharing the article, I’ll dive into it.
I’m linking the article which compares various autodiff methods for differential equations in Julia, with some examples from the PKPD litterature: https://arxiv.org/pdf/1812.01892.pdf.
Ideally, Stan would let the user choose which differentiation method to use (forward, adjoint, and maybe more). @bbbales2, I remember you were working on implementing the adjoint method in Stan and per this last thread, things seemed to be moving forward pretty smoothly: Adjoint task force for ODEs Are there any updates on this?
@wds15 It makes sense that with a small number of parameters, K, but a large number of states, N, the adjoint method outperforms forward sensitivity, given they respectively solve 2N +K and N + NK states. Figure 2 in the paper I linked (Rackauckas et al, 2018) examine a 2 state PDE, which is why you need \sim10^{2.5} parameters before you get any gain from the adjoint method. With 47 states, yes, it’s a different story.