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This paper establishes a unifying theory for on-device gradient computation in physical computing systems, addressing the challenges posed by the model-reality gap in gradient-based optimization. By employing the adjoint method, the authors delineate sufficient conditions for linear and nonlinear systems to compute exact gradients using their own hardware, revealing that linear systems can utilize damping or gain while maintaining reciprocity, and nonlinear systems require specific conditions like time-reversal symmetry. The findings not only recover existing methodologies such as Equilibrium Propagation and Hamiltonian echo backpropagation but also extend the framework to a broader class of non-Hermitian systems, thereby providing a comprehensive foundation for future physical learning algorithms.
Exact on-device gradient computation is achievable across a range of physical systems, challenging the limitations of traditional digital twin approaches.
Physical computing systems exploit device dynamics for computation, but their gradient-based optimization is challenging: backpropagation through a digital twin suffers from model-reality gap. On-device gradient computation could resolve this issue, and a handful of theoretical and experimental studies have proposed ways to achieve it. Yet a unifying theory identifying when a physical system can compute the gradient of its own performance has been missing. Here we develop such a unification, based on the adjoint method: we identify sufficient conditions under which the adjoint field required for formally exact gradients can be generated on the same hardware that performs the computation. Linear and nonlinear systems obey fundamentally different conditions: for linear systems damping or gain is admissible provided reciprocity is preserved. For nonlinear trajectory systems the sufficient conditions are reciprocity of the linearized system and the existence of a time-reversal mirror. Algorithmically, the nonlinear case requires infinitesimal nudging, whereas linear systems admit a finite-amplitude experiment. We recover Equilibrium Propagation, Hamiltonian echo backpropagation, fully forward mode training and in situ gradient methods in integrated-photonic and free-space-optical systems. We further show that reciprocity is only the simplest instance of a more general intertwining condition, which extends exact on-device gradient computation to a class of non-Hermitian, non-reciprocal systems. Further generalizations include time-dependent parameters, Onsager-reciprocal dynamics and nonlinear, PT-symmetric Schr\"odinger equations. Our work provides a unified theoretical basis for formally exact physical learning algorithms and a template for constructing them across a range of physical systems.
