🗻 autograd: broadband objectives with single adjoint simulation

[tylerflex]

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[tylerflex]

@momchil-flex @yaugenst-flex sorry for the "bait and switch". I just realized ultimately it makes more sense to unify the multi-frequency under one single approach rather than have various paths. So what I'm doing now is this:

The vip-containing SimulationData gives us a list of a bunch of adjoint sources. with SourceTime amplitudes corresponding to the amplitudes needed (assuming that the adjoint Simulation is normalized by that source time).

I sort all of the adjoint sources by source spatial dependence (eg. Source without the time dependence). Since the fitting approach only needs to be done source by source.

I then compute the coupling through source_time.spectrum() between all of the adjoint frequencies. This tells me how much injecting unit amplitude at frequency i affects frequency j.

Then I solve a linear system with least squares to find a new set of amplitudes to inject with each source. I also turn off normalize_index in the adjoint simulation, since this is taken care of in these amplitudes already`.

To avoid numerical issues, after collecting all of the adjoint sources, I normalize them with the np.linalg.norm() of the combined. This way, we always inject power with unit of amplitude 1 into the adjoint simulation and avoid divergence or really low field values (that potentially underflow?)

After the adjoint sim is run, the adjoint fields are multiplied by this norm() to get the expected values again.

Note: I have NOT implemented yet:

1. automatic fwidth reduction, if the fit is bad.
2. automatic run_time setting in adjoint source if (1) or if the adjoint source spectra combine to create something that lasts a long time.

I'll continue testing next week, and when it seems ready will release it for review, thanks!

[tylerflex]

OK This PR is ready for review again.

Here's how it handles multi-frequency adjoint problems:

if the number of frequencies in the objective function is 1, it just does what we did before.

if the number of "ports" in the objective function (to put another way, if we can formulate a broadband adjoint source with one source). We do broadband adjoint with a single source and then post-normalize the adjoint fields using the vjp values.

Otherwise, we would use other more complicated approaches, in this case, I raised a NotImplementedError.

I'm still writing the two demos

1. WDM
2. bandpass filter.

Should be done tomorrow or thursday.

[yaugenst-flex]

Cool not much to say other than the comments that I left!

[momchil-flex]

So you are excluding the SourceTime of all adjoint sources, and counting their unique definitions after that exclusion?

In some cases, like when differentiating through FieldData.flux, the adjoint source would be a CustomFieldSource, right? Does this work for those too?

[tylerflex]

So you are excluding the SourceTime of all adjoint sources, and counting their unique definitions after that exclusion?

That's right.

In some cases, like when differentiating through FieldData.flux, the adjoint source would be a CustomFieldSource, right? Does this work for those too?

Good question. I'm not really sure to be honest. Right now it would treat two otherwise identical CustomFieldSource with different data as separate "ports". I think this is the proper handling though, since the data in the source defines the spatial dependence of J(r, ω) and we're trying to do here is essentially separate J(r) and J(ω), so two different datasets should be treated as different J(r) I think.

The only case I think this could become slightly ambiguous for the user is if they had two identical FieldMonitor "a, and b and then for some reason a and b had different contributions to the objective function. In that case, we could come up with a scenario where the adjoint sources for a and b could theoretically be combined into one (ie by just summing the datasets with the corresponding source time amplitudes). But this seems like such an edge case that it's probably not worrying about to me.

[tylerflex]

just to clarify, in the case described above, we would simply require a separate adjoint simulation for a and b, but it wouldn't error in an unexpected way.

[momchil-flex]

My worry was rather that I'm still not sure what happens if you look at the json only, i.e. if the actual data gets compared or just a dummy placeholder tag or something that would be identical for all custom sources. If it works the way you describe it sounds good.

[tylerflex]

Hm.. actually the way it is set up, I suppose the datasets would be excluded from the jsons. So two identical FieldMonitors a and b would actually be considered the same spatial dependent AdjointSource, and therefore would potentially confuse the broadband adjoint. I'll probably need to look into testing this as an edge case. but for otherwise different FieldMonitor definitions, it will have no problem.

[momchil-flex]

Yeah, that's the edge case I had in mind. Although... technically if they're identical in their definitions, the recorded datasets should also be the same?

Ah but something weird may happen where the two are used in different ways in the objective function and so the adjoint source currents are different?

Very edgy.

[tylerflex]

well the recorded datasets would be the same, but what we get back is the VJP not the recorded datasets. so yea if the data is used in two different ways, it wouldn't show up.

I think one way to distinguish spatial dependence this could be to instead make a copy of all of the adjoint sources with the same source time, and then hash these. But then I dont know exactly if the datasets are included in the hash or not..

[tylerflex]

yea so I gave this a stab (hashing temporary copies of the sources with identical source_time) and it seemed to work well

93e3b31

This should be good enough.