In-Situ Adjoint Optimization Protocol for a Class of Nonlinear Parity-Time Symmetric Systems

Gradient-based optimization underpins many cyber-physical systems, yet digital implementations suffer from latency and power cost. Physical systems offer a path to drastically lower both deficiencies by embedding gradient sensitivity analysis in hardware. Such an embedding may be facilitated by the Adjoint Equations, which is able to extract the sensitivities of all system parameters simultaneously. However, physically realizing the adjoint equations is non-trivial, specifically for nonlinear systems that violate time-reversal symmetry. In these cases, the conventional adjoint equations differ from the forward equations and may not even be physically realizable, posing a major challenge. We have identified a class of nonlinear parity-time (PT)-symmetric platforms that overcome this challenge and allow us to reformulate the adjoint, so it is identical to the forward system, enabling a single device to perform both computations via judiciously tailored excitations. The required drive signals exploit spatial, temporal, and phase degrees of freedom inherent to wave transport. We demonstrate the protocol in-silico using coupled-mode theory modeling and discuss implementations to various physical systems, pointing to low-latency, low-power optimization on physical hardware.