In-situ adjoint protocols for nonlinear PT-symmetric self-optimizing machines
Adjoint methods provide a powerful route for gradient-based optimization, but their physical implementation is obstructed in generic nonlinear systems because the adjoint dynamics requires backward-time evolution, Jacobian transposition, and terminal-value constraints. Here we show that nonlinear parity-time ($\mathcal{PT}$)-symmetric systems overcome this obstruction. Using a class of nonlinear non-Hermitian resonator networks, we establish symmetry relations that map the formal adjoint dynamics onto experimentally accessible forward-time evolutions supplemented by controlled injections. This construction enables exact in-situ evaluation of adjoint gradients without requiring explicit backward propagation or matrix transposition. We demonstrate the approach in nonlinear $\mathcal{PT}$-symmetric resonator chains, where the resulting optimization protocol autonomously discovers parameter configurations that realize prescribed spatio-temporal functionalities, including uniform energy redistribution and targeted wave transport at predefined time windows. Our results identify $\mathcal{PT}$ symmetry as a resource for implementing computational sensitivities within physical systems and establish a route toward self-optimizing nonlinear machines.