Physical Adjoint Learning in Wave Systems with Asymmetric Couplings

Developing self-optimizing physical systems that learn directly from their response to external stimuli offers a promising route toward adaptive and intelligent machines. A key ingredient is the ability to evaluate optimization gradients within the physical hardware, bypassing the need for accurate numerical models. In wave systems, \textit{in-situ} adjoint methods provide this capability by exploiting reciprocity to physically realize the adjoint dynamics. However, this approach generally breaks down in nonreciprocal systems, where the adjoint field cannot be directly accessed through the original device. Here, we develop an \textit{in-situ} adjoint optimization framework that extends physical gradient computation to wave systems with asymmetric couplings through computationally low-cost similarity transformations between the forward and transposed operators. We utilize available degrees of freedom like tailored injected excitations, targeted measurements, and tunable parameters, to enable adjoint field reconstruction through forward experiments on the same physical platform. We demonstrate the versatility of the protocol through broadband transmission and isolation, coherent perfect absorption, programmable wavefront shaping, and spatially selective sensing. Our results provide a route to self-optimization in complex nonreciprocal wave systems through efficient physical gradient-based learning. The number of experiments required by the protocol does not increase with the number of optimization parameters, and the method remains applicable even when parts of the system are completely inaccessible.