Let's use gradient descent to optimise the "refractive index" at each point in a square in the middle. After a few iterations we get the following (grey) refractive index leading to the following simulation. Note how it all conspires to arrive at the right point...
You could solve this problem using automatic differentiation. It's a lot like training a convolutional neural net because a single time step here is essentially a (spatially varying) convolution and you're optimising the parameters that determine a long sequence of such convolutions...
In this example I think (not 100% sure) the optimization has tried to construct elliptical mirrors with the start and end at the foci to focus the "direct" wave and parabolic mirrors at the ends to focus the waves that wrap around the domain...
But this is much the same problem seismologists solve - finding the material properties based on what signals arrive where from a source. And seismologists were doing this long before convolutional neural nets were invented. So I thought I'd try something closer to what they do. It's also more insightful than just using AD which is something of a black box.
Anyway, here's the result of solving the same problem but getting the focus to happen much later. There's a sort of fractally refractive index now...
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