Introducing Adjoint Solver 2.0

[smartalecH]

This is an initial stab at the new and improved adjoint solver. This update is meant to enable an adjoint solver that is simple, flexible, and accurate.

Major differences and features from previous version:

• Support for broadband dispersive materials
• Calculates gradients for multiple frequency points simultaneously -- even if the objective function has multiple arguments
• Plug and play support for filters, projections, "robust" implementations, etc.
• Automated differentiation of objective functions using autograd
• Accurate gradients rigorously benchmarked against their finite differences
• Simplified API with minimal code overhead
• Built-in bilinear interpolation basis

The quickest and easiest way to get up to speed with the new features is to review the accompanying notebooks:

• Introduction
• Waveguide bend optimization
• Filters and projections
• Broadband simulations

I'm sure several changes will have to be made before merging. Some of those include:

• Docs
• Tests
• Fix interpolation gradient with lots of design parameters per pixel
• Supply typical filters as built-in functions.
• Specify a minimum run time, if you need to force it to run longer.
• Pass an freq array (like other features recently added).
• Instead of passing a single point to determine when to stop the simulation, look at the objective arguments and/or the Fourier transforms of the fields in the design region.
• Maybe call the variable design_variables = [...] instead of basis = [...].
• Would be good to support passing a list of objective functions (all with the same arguments).
• Add a tutorial that designs a fabricatable structure (using a simple design like the bend)
• Extend a basic tutorial (like the phase grating) to calculate the sensitivity
• Simple 3d example

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[stevengj]

The bend optimization results look like they are converging towards:

[stevengj]

Comments:

• Supply typical filters as built-in functions.
• Specify a minimum run time, if you need to force it to run longer.
• Pass an freq array (like other features recently added).
• Instead of passing a single point to determine when to stop the simulation, look at the objective arguments and/or the Fourier transforms of the fields in the design region.
• Maybe call the variable design_variables = [...] instead of basis = [...].

[stevengj]

Would be good to support passing a list of objective functions (all with the same arguments). Then you do one forward simulation to get all of the objective functions (at each frequency), but then do a separate adjoint simulation for each objective function.

The reason to do this is to support arbitrary constrained optimization problems. For example, suppose we are maximizing power in mode 1 with a constraint that the power in mode 2 is ≤ 0.1 × (power in mode 1).

(We could still support passing a single objective function as now, as a special case.)

[stevengj]

If you plot the gradient of an optimized structure, it should be positive where ε=12 and negative where ε=1 (or vice versa, depending on whether you are maximizing or minimizing), and nearly zero at points where ε takes on intermediate values. (This is just the KKT conditions.)

[smartalecH]

I've made the requested changes but discovered an annoying bug with my interpolator. It seems that if I have far more design parameters than meep pixels in the design region, the gradient doesn't match up with its finite difference approximate.

You can view this effect in this notebook.

Interestingly I can still optimize fairly well (as demonstrated in the filtering example).

[smartalecH]

Tutorials are fixed.

Filtered Waveguide Bend designs a fully binarized, broadband bend with a minimum length scale constraint:

with a decent response:

I'd say it's ready for merging. I can pull docs and "official" tests together in another PR.

[franklee90]

Hi, I tried to run the introduction script but got this error:

AttributeError: module 'meep.adjoint' has no attribute 'BilinearInterpolationBasis'

I updated my meep using the simpetus nightly build before trying. Has the adjoint solver 2.0 been included in it?

[stevengj]

No, it was just merged, so it will take 24 hours or so to appear in the nightly builds.