  When solving, modelling 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 prospect 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. Although, the distributed nature of photonic platforms
leads to the absence of an equivalent approximation to Kirchhoffs law, the main
principle used for representing physical systems using circuits. Here we argue
that in absence of a straightforward parallelism and homomorphism can be
induced. We introduce a photonic platform capable of mimicking Kirchhoffs law
in photonics and used as node of a finite difference mesh for solving partial
differential equation using monochromatic light in the telecommunication
wavelength. We experimentally demonstrate generating in one-shot discrete
solutions of a Laplace partial differential equation, with an accuracy above
95% relative to commercial solvers, for an arbitrary set of boundary
conditions. Our photonic engine can provide a route to achieve chip-scale, fast
(10s of ps), and integrable reprogrammable accelerators for the next generation
hybrid high performance computing.
