Unifying Physical Backpropagation

Abstract:Physical computing systems exploit device dynamics for computation, but their gradient-based optimization is challenging: backpropagation through a digital twin suffers from a 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 (quantum) Equilibrium Propagation, Hamiltonian echo backpropagation, fully forward mode training and in situ gradient methods in integrated-photonic and free-space-optical systems. Finally, we show that reciprocity is a special case of more general intertwining conditions. For linear systems, these permit exact on-device gradients in a class of non-Hermitian, non-reciprocal systems. For nonlinear trajectories, they combine with generalized time-reversal mirrors to cover, e.g., PT-symmetric equations. The framework also includes time-dependent parameters and Onsager-reciprocal dynamics, providing a unified basis for formally exact physical learning.

Disordered Systems and Neural Networks (cond-mat.dis-nn); Emerging Technologies (cs.ET); Machine Learning (cs.LG); Optics (physics.optics)

arXiv:2608.11585 [cond-mat.dis-nn]

(or arXiv:2608.11585v2 [cond-mat.dis-nn] for this version)

https://doi.org/10.48550/arXiv.2608.11585 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.