Fig. 2: Solving integral equations with the Miller architecture.

a Continuous distribution of the kernel K(u, v). The black dots denote the spatial points where the kernel is sampled to derive its 7 × 7 discrete version K reported in the color plot (b). The maximum amplitude of the complex value in the color circle insets is set to 1.15 and (2/7)(1.15) in a and b, respectively. c Schematic of flow through the MZI network wherein the color of the arrows denotes the complex transmission between the MZI ports with a maximum value of 1. d, e The extracted solution (real-red and imaginary-yellow) of the integral equation (1) with the proposed kernel under two different real inputs (blue). d The input is a constant value 1, while in e, it is an appropriately sampled Gaussian distribution centered on u =−2/7. This solution is evaluated in three different ways: the highly sampled (n = 1001) true solution (continuous line) evaluated using linear algebra (i.e., \({{{\bf{x}}}}={\left({{{\bf{I}}}}-{{{\bf{K}}}}\right)}^{-1}{{{\bf{c}}}}\)) method, the solution (crosses) with n = 7 also calculated directly using linear algebra, and the solution (circles) calculated via the simulation of our MZI mesh. All three show excellent agreement