Patrick Kidger

Take a look at https://arxiv.org/abs/2106.11028 and https://docs.kidger.site/diffrax/api/interpolation/. These discuss handling causality in the input. (It's quite complicated, unfortunately!)

- Hard to comment on out-of-sample performance I'm afraid. - For the values for `t=...`, this is a problem that is substantially simpler when using Diffrax. This allows you to...

Hey there! Thank you, I'm glad you like my work. For integrating to a steady state -- indeed, it's possible to do much better than simply set a large number....

What might help - we have an implementation [here](https://github.com/google-research/torchsde/blob/5fa77533a3081ebc2f5f5de5a594111081d09b0f/torchsde/_brownian/brownian_interval.py), which includes a few fun things we didn't mention in the paper, like Levy area approximation via Davie-Foster approximations.

There are definite similarities between the two approaches. Technical point: there are two slightly different methods, Davie and Davie-Foster. What I say basically applies to both. KPW approximates Brownian motion...

Mathematically, pretty much yes. (Although the point of the Brownian Interval, over the Brownian Tree, is that it isn't restricted to using just midpoints.) The concern is computational -- memory...

It's an interesting paper but I don't think I see the relevance to this discussion.

That is correct. Whether or not that's an issue depends on whether you actually need your solution to be deterministic wrt just the global entropy, and whether you are using...

Returning to this: Equinox version 0.10.0 should include some performance improvements for `eqx.filter_pmap`. I'd be curious to know this helps out here.

That all sounds reasonable to me! I'd happily accept a PR on these. (I probably wouldn't bother making reduce public though, that's a pretty small helper function.)