  Solving mathematical equations faster and more efficiently has been a Holy
Grail for centuries for scientists and engineers across all disciplines. While
electronic digital circuits have revolutionized equation solving in recent
decades, it has become apparent that performance gains from brute-force
approaches of compute-solvers are quickly saturating over time. Instead,
paradigms that leverage the universes natural tendency to minimize a systems
free energy, such as annealers or Ising Machines, are being sought after due to
favorable complexity scaling. Here we introduce a programmable analog solver
leveraging the mathematical formal equivalence between Maxwells equations and
photonic circuitry. It features a mesh network of nanophotonic beams to find
solutions to partial differential equations. As an example, we designed,
fabricated, and demonstrated a novel application-specific photonic integrated
circuit comprised of electro-optically reconfigurable nodes, and experimentally
validated 90% accuracy with respect to a commercial solver. Finally, we tested
this photonic integrated chip performance by simulating thermal diffusion on a
spacecrafts heat shield during re-entry to a planets atmosphere. The
programmable light-circuitry presented herein offers a facile route for solving
complex problems and thus will have profound potential applications across many
scientific and engineering fields.
