Induced homomorphism: Kirchhoff’s law in photonics

When solving, modeling or reasoning about complex problems, it is usually convenient to use the knowledge of a parallel physical system for representing it. This is the case of lumped-circuit abstraction, which can be used for representing mechanical and acoustic systems, thermal and heat-diffusion problems and in general partial differential equations. Integrated photonic platforms hold the prospective to perform signal processing and analog computing inherently, by mapping into hardware specific operations which relies on the wave-nature of their signals, without trusting on logic gates and digital states like electronics. Here, we argue that in absence of a straightforward parallelism a homomorphism can be induced. We introduce a photonic platform capable of mimicking Kirchhoff’s law in photonics and used as node of a finite difference mesh for solving partial differential equation using monochromatic light in the telecommunication wavelength. Our approach experimentally demonstrates an arbitrary set of boundary conditions, generating a one-shot discrete solution of a Laplace partial differential equation, with an accuracy above 95% with respect to commercial solvers. Our photonic engine can provide a route to achieve chip-scale, fast (10 s of ps), and integrable reprogrammable accelerators for the next generation hybrid high-performance computing.